Morphing of an adaptive shock control bump using pressurized chambers

Abstract

The dawn of research on shock and boundary layer interaction control dates back to the 1970s, when humped transonic aerofoils were first studied as a means to improve the performance of supercritical aerofoil technology at off-design conditions. Since then, shock control bumps have been found to be promising devices for such kind of flow control. They have a smearing effect on the shock wave structure achieved through isentropic pre-compression of the flow upstream of the main shock and can significantly lower wave drag without incurring unacceptable viscous losses. However, their performance is strongly dependent on a set of geometrical parameters which must be adjusted according to the ever-changing flight conditions. A concept for an adaptive shock control bump is therefore presented. The proposed actuation mechanism aims at a compact, lightweight and simple structure which could be integrated into the spoiler region of near-future aircraft without major design changes required. Numerical optimization of a simplified analytical model of the structure is used to investigate the shock control bump adaptation to various aerodynamic target shapes. Compromises between geometrical conformity and both structural and actuation related requirements are studied. Furthermore, an outlook is given on design issues related to three-dimensional effects on a finite span shock control bump.

Keywords

shock control bump wave drag minimization morphing spoiler pressure actuated adaptive structure structural optimization

1. Introduction

Aircraft manufacturers are aware that even small efficiency gains can represent substantial savings for air operators in the ever more competitive air transport sector. Therefore, modern aircraft technologies are focusing on the reduction of direct operating costs and increase in mission flexibility as a means to achieve a competitive advantage. Most of the high economic and industrial impact technologies are dependent on the optimization of the aerodynamic efficiency of the main wing which has long been an area of design compromises for the engineer: a narrower cruise envelope leads to higher efficiency gains with worse off-design performance, while a wider cruise envelope means less efficiency gains within the operating conditions.

Large scale morphing technologies dealing with the adaptation of the entire wing as described by Smith et al. (1992) ensure that each operating point is a design optimum. Nevertheless, this approach requires complex actuation mechanisms and can be seen as a long-term solution. A more feasible approach would be adapting the aerofoil geometry by focusing on individual movable parts, that is, leading edge, flaps and spoiler devices. In fact, the EUROSHOCK I and II programs showed that the use of a physical bump in order to adapt a relatively small portion of the suction side of the main wing, as initially proposed by Ashill and Lock (1992), is an effective way of minimizing wave drag due to the strong shock waves that develop at transonic cruise conditions. Thereafter, the so-called shock control bump (SCB) has been subject to various aerodynamic investigations and its role as a flow control device is becoming increasingly popular.

This study focuses on the structural realization and actuation of such devices. It is therefore useful to first introduce basic aerodynamic principles concerning SCBs in order to understand how they relate to structural constraints.

2. Aerodynamic principles

Increasing the free-stream Mach number or angle of incidence of a transonic aerofoil will lead to the development of a supersonic region on the upper surface. In turbulent aerofoils, the supersonic flow will usually be terminated by an isentropic compression or a weak shock wave, as shown in Figure 1(T). If either free-stream parameter is increased, commonly the shock will strengthen leading to extra wave drag culminating in separation which defines the drag-rise and buffet boundaries. For a laminar aerofoil –Figure 1(L)– the flow undergoes a continuous acceleration on the upper surface causing stronger shock waves, even for the design conditions.

Across the shock, entropy increases and total pressure decreases. Wave drag is then generated and it impacts cruise efficiency of current aircraft configurations.

SCBs can split the stronger single shock into a series of weaker oblique shocks or compression waves that decelerate the flow more isentropically than the uncontrolled shock wave. The result is a reduced stagnation pressure loss across the shock leading to lower drag. Figure 2 illustrates the principle of operation of SCBs in transonic flight.

SCB geometry is usually comprised of a ramp upstream of the nominal shock position, a short crest region and a tail. The function of the ramp is to generate an oblique shock or multiple oblique compression waves ahead of the main shock deflecting the incoming supersonic flow away from the surface. Close to the crest, a near-normal shock decelerates the flow to subsonic velocities. The tail then brings the post-shock flow back to the aerofoil surface.

SCBs can be classified as two-dimensional (2D) SCBs – a constant profile along the wing span and three-dimensional (3D) SCBs – a series of short-width bumps distributed along the wing. This work will focus on 2D SCBs as they present the highest potential for efficiency gains.

Figure 3 summarizes the characteristic geometric parameters of an SCB: the bump is located on an aerofoil surface of chord c at a position $x_{0}$ from the leading edge. It has a crest of length $c_{B}$ and height $h_{B}$ , which is schematically represented by point $C (c_{B}, h_{B})$ . The bump is also defined by its length $l_{B}$ and its shape $f_{B}$ .

Table 1 shows reference ranges considered by a multitude of aerodynamic studies on SCBs. A significant aspect to retain is the fact that SCB research has essentially focused on bumps with heights of the order of the local boundary layer thickness. In addition, typical peak shock strength is around $M = 1.3 \pm 0.2$ .

Table 1.

Characteristic SCB parameters and typical values investigated.

Symbol	Parameter	Typical values
$h_{B}$	SCB height	$δ_{0} \pm δ_{0}$
$l_{B}$	SCB length	$\frac{l_{B}}{c} \in [12 %; 20 %]$
$c_{B}$	Crest length	$\frac{c_{B}}{l_{B}} \in [30 %; 80 %]$

SCB: shock control bump.

2.1. Aerodynamic performance of SCBs

Previous works by Dargel et al. (1999) and Sommerer et al. (2000) have shown that the drag reduction potential of SCBs depends strongly on the correct bump position, height and location of the bump maximum. The bump height depends on the actual flight conditions and the aerofoil geometry, but can be estimated to be within 0.5% of the local chord length. An interesting result from these investigations is the fact that the detailed bump shape does not have a strong effect on the bump effectiveness.

Figure 4 is helpful in understanding why the bump position, height and location of the bump maximum play a major role in SCB performance. When the shock is at the optimum location –Figure 4(b)– a ‘cleaner pressure’ rise has the least negative effect on the boundary layer. The off-design cases –Figure 4(a) and (c)– introduce a re-expansion and secondary shock structures which significantly impact the incoming boundary layer: when the shock is upstream of the optimum location, the bump accelerates the flow forming a secondary shock –Figure 4(a); when the shock is downstream of the optimum location, there is an unfavourable expansion over the bump crest leading to a stronger shock and ultimately to boundary layer separation –Figure 4(c).

As a consequence, static SCBs present a narrow region of effectiveness. When the flight conditions lead to non-optimum shock locations, SCB performance is worse than that of a clean aerofoil. Since there is some shock mobility during cruise, it is essential to be able to continuously adapt the significant SCB parameters according to the actual shock position. This sets the case for an adaptive SCB, where the main actuation objective is the ability to adjust the crest position.

3. Structural concept, model and optimization

While there are numerous publications dealing with the aerodynamic design of SCBs, the structural realization of such a structure system is still a key challenge. For current transport aircraft, even though during flight shock movement is always an obstacle, the shock location is generally restricted to the rear portion of the wing, that is, the region closer to the spoilers and the wing trailing edge. In order to allow an easy integration of SCBs in near-future transport aircraft, the proposed system location is the spoiler region. The available design space is therefore extremely limited and the main drivers for concept selection should be a compact design, lightweight structure and low complexity actuation mechanism for the SCB deployment.

3.1. Concept

The main design objective is to achieve a structure capable of conforming to prescribed optimal aerodynamic bump shapes (or at least a set of crest positions) while having enough structural stiffness in order to resist aerodynamic loads. Therefore, a new integral spoiler structure has been selected.

The new integral spoiler structure (Figure 5) is comprised of three layers: the top layer is an active surface where the SCB is generated, the bottom layer is a load-carrying structure which supports both aerodynamic loads and actuation loads for the SCB and the middle layer is where the internal actuation mechanism for the SCB deployment is fitted. This approach does not require a complete redesign of the spoiler deployment mechanism since the actuation of the SCB is independent of the spoiler deployment mechanism. The passive structure greatly simplifies the design of the SCB actuation system as it provides the necessary support for the actuation loads. This design approach separates the functional (active) structure from the supporting structure thanks to a mechanical parallel actuation system as Wadehn et al. (2002) and Bein et al. (2000) proposed. These authors also proposed the use of shape-memory alloy (SMA) actuated deformation and pressurized tube springs as variations on the linear actuator for internal actuation. However, this approach requires the use of numerous actuators.

A spoiler concept using pressurized chambers was selected as a pioneering actuation technology that is able to combine the key requirements. The SCB deployment and its adaptation to the changing flight conditions is realized by differential pressurization of chambers inside the spoiler body, as shown in Figure 6. A series of chambers could be combined into one single part during manufacturing. This reduces the actuation mechanism to a sealed assembly that can easily be replaced in case of a repair. The chambers are supported by the conventional load-carrying structure and spanwise stiffeners can be designed according to the needs of the parallel actuation system using stringers embedded into the interfaces (chamber walls) between each chamber. Therefore, the cross-sectional stringer shape must be designed to allow deformation in the direction of the chamber height while at the same time providing spanwise support. Since pressure acts uniformly across the entire span of the spoiler, spanwise waviness issues are not expected. Moreover, the elastomeric material of the chamber walls allows for a compact and comparably lightweight mechanism. Finally, since pressure is allowed to act on the entire portion of the flexible skin that contacts each chamber, it is predictable that the number of chambers, that is, actuators, might be reduced when compared to previous concepts, also contributing to a more compact and lightweight structure.

3.2. Analytical model

A design tool was developed to provide a systematic way of investigating key structural and functional interdependencies related to the actuation of the selected design concept. This tool aims to provide the foundations for an accurate sizing of the functional layer of the SCB and give an insight on the trade-offs between structural integrity and geometrical conformity qualities. The simplified analytical model used by the design tool is now presented.

Figure 7 shows the simplified spoiler model where the x and y coordinate directions are the chordwise and spanwise directions, respectively. The upper spoiler skin is modelled as a transversely loaded plate, while the main lower body is assumed to be comparably much stiffer. The upper skin is clamped to the support structure and free at the spoiler tips. Given that the considered SCB height-to-length ratios are below 5%, small strains and moderate rotations are expected. For this reason, the Kirchhoff plate theory will be used. For transversely loaded Kirchhoff plates, the static bending equation is

\begin{matrix} D_{11} \frac{\partial^{4} w_{0}}{\partial x^{4}} + 2 (D_{12} + 2 D_{66}) \frac{\partial^{4} w_{0}}{\partial x^{2} \partial y^{2}} \\ + D_{22} \frac{\partial^{4} w_{0}}{\partial y^{4}} = q (x, y) \end{matrix}

(1)

where $w_{0}$ is the transverse displacement of a point on the plate midplane, $D_{ij}$ are the bending stiffnesses and q is the distributed transverse loading. Considering a reference frame centred at the spoiler leading edge, the boundary conditions for the clamped edges are given by

Γ_{u} : w_{0} = 0, {\frac{\partial w_{0}}{\partial n} |}_{n = x} = 0, x = {0, l_{B}}

(2)

where $Γ_{u}$ indicates the essential boundary conditions, n indicates the direction normal to the clamped edge and $l_{B}$ indicates the bump length. For the free edges, natural boundary conditions are defined using the bending moment $M_{nn}$ and the Kirchhoff free-edge condition

Γ_{σ} : M_{n n} = 0 V_{n} \equiv Q_{n} + \frac{\partial M_{n s}}{\partial s} = 0, y = \pm \frac{s_{s p}}{2}

(3)

where $V_{n}$ is the total vertical force given in terms of the vertical resultant $Q_{n}$ and moment $M_{ns}$ and $s_{sp}$ is the spoiler span. The first index represents the normal direction to the edge surface. For the moments, the second index is the direction along which the force causing the moment is acting. Once more, n is the normal to the free edge, that is, ${\hat{e}}_{y}$ and s is the tangent direction to the free edge, that is, ${\hat{e}}_{x}$ .

The materials investigated will be restricted to single layer specially orthotropic and isotropic laminates. For these cases, the bending stiffnesses, vertical resultant and moments can be written in terms of the lamina engineering constants and equation (1) can be recasted into

\begin{matrix} \frac{\partial^{4} w_{0}}{\partial x^{4}} + 2 [ν_{21} + 2 \frac{G_{12}}{E_{1}} (1 - ν_{12} ν_{21})] \frac{\partial^{4} w_{0}}{\partial x^{2} \partial y^{2}} \\ + \frac{ν_{21}}{ν_{12}} \frac{\partial^{4} w_{0}}{\partial y^{4}} = \frac{q}{\hat{D}}, \hat{D} \equiv \frac{E_{1} h^{3}}{12 (1 - ν_{12} ν_{21})} \end{matrix}

(4)

where h is the laminate thickness. The boundary conditions can also be written in terms of the displacement $w_{0}$ . For the natural boundary conditions, laminate constitutive equations for specially orthotropic materials have been used to resolve the forces and moments in equation (3) into the displacement $w_{0}$

\begin{matrix} Γ_{u} : w_{0} = 0, \frac{\partial w_{0}}{\partial x} = 0, x = {0, l_{B}} \\ Γ_{σ} : ν_{12} \frac{\partial^{2} w_{0}}{\partial x^{2}} + \frac{\partial^{2} w_{0}}{\partial y^{2}} = 0 \\ (ν_{12} + 4 \frac{G_{12}}{E_{2}} (1 - ν_{12} ν_{21})) \frac{\partial^{3} w_{0}}{\partial x^{2} \partial y} \\ + \frac{\partial^{3} w_{0}}{\partial y^{3}} = 0, y = \pm \frac{s_{sp}}{2} \end{matrix}

(5)

The problem requires approximate analytical solution methods like the Ritz method or numerical methods like the finite element method (FEM). However, if the SCB span to length ratio is large enough, and assuming that the pressurization of the chambers is sufficiently uniform, $y >> x$ and $q \approx q (x)$ . Under these circumstances, the central portion of the active layer behaves as a long semi-infinite plate strip and the cylindrical bending approximation greatly simplifies the solution of equation (4) by reducing the problem to one dimension. The $\partial / \partial y$ derivatives vanish and the natural boundary conditions are eliminated, which means $w_{0} = w_{0} (x)$ . The governing equation and boundary conditions are simply

\frac{d^{4} w_{0}}{d x^{4}} = \frac{q}{\hat{D}}, \hat{D} \equiv \frac{E_{1} h^{3}}{12 (1 - ν_{12} ν_{21})}

(6)

Γ_{u} : w_{0} = 0, \frac{d w_{0}}{dx} = 0, x = {0, l_{B}}

(7)

One should note the similarity between equation (6) and the classical beam bending equation. The difference is in the bending stiffness under cylindrical bending which includes the Poisson’s ratios due to the plane strain assumption. Figure 7 also shows the area where the cylindrical bending approximation is valid in red.

For the SCBs investigated within the Low Drag Aircraft in Operation (LDAinOp) program, the SCB span-to-length ratio, that is, aspect ratio (AR), is often larger than seven. Even for lower aspect ratio SCBs located in the inner region of the main wing ( $AR > 2.5$ ), the approximation is reasonable for approximately 80% of the SCB span. In the outer region near the spoiler tips, there is a complex spanwise deformation.

Figure 8 shows the effect of SCB AR on the spanwise deformation of the SCB crest (located at $x = c_{B} = l_{B} / 2$ ) in terms of the dimensionless spanwise coordinate $\tilde{y} = y / s_{sp}$ . Each curve is associated with a different AR for the chosen pressurization and was obtained using a Mindlin–Reissner FEM model of the SCB. The topology of the spanwise deformation at other chordwise positions of the active layer is similar to that shown in Figure 8, although the deflections involved are smaller. As the AR increases, a region of uniform spanwise deformation appears. The deflection is accurately predicted by the aforementioned approximation.

Figure 9 shows the relative difference between the deflection of the half-span crest using the aforementioned FEM model and the cylindrical bending approximation (solid line). Due to the linearity of equation (6), the results are similar for different pressurization levels as shown.

The dashed line represents the relative difference between the mean deflection of 80% of the spoiler span and the cylindrical bending approximation. In both cases, for $AR > 2$ , the error of the approximation remains below 0.5% and for that reason it was deemed an adequate simplification that allows an efficient structural solver and avoids a full plate model.

3.3. Structural module

A structural module was developed based on the analytic element method (AEM) approach as described by Policarpo and Matos Neves (2012). This allows for a structural solver which is both accurate and efficient. The module is then used in an optimization environment where the desired bump shapes are fed to the optimization algorithm through an objective function as target shapes in order to find the optimized design parameters that minimize the difference between the target shape (i.e. the desired aerodynamic bump shape) and the optimized shape.

Figure 10 shows the active surface of the SCB idealized as a series of plate strips spanning along the y direction which are actuated by the pressurized chambers inside the spoiler body. A spring is used to model the stiffness of the interface between consecutive chambers whose pressure is kept uniform.

As a first approach, the pressurization of each chamber will be assumed uniform in the spanwise and chordwise directions. The deflection of a point in the midplane of the plate strip actuated by the ith chamber is obtained from equation (6)

w_{0_{i}} (x) = \frac{p_{i}}{24 {\hat{D}}_{i}} x^{4} + C_{1}^{i} x^{3} + C_{2}^{i} x^{2} + C_{3}^{i} x + C_{4}^{i}

(8)

where ${\hat{D}}_{i}$ is the bending stiffness of the ith plate strip and $C^{i}$ is the set of constants to be determined using the boundary conditions. Figure 11 shows an element of plate strip under cylindrical bending and the sign convention for the transverse force ( $Q_{x}$ ) and bending moment ( $M_{xx}$ ), as well as for the nodal degrees of freedom (DoFs) (i.e. the nodal deflections and rotations).

Using the definitions of the transverse force $Q_{x}$ and bending moment $M_{xx}$ and the strain–displacement relations, the loads on the left (I) and right (J) nodes of the ith plate strip can then be related to the $C^{i}$ analytic constants

\begin{matrix} {\begin{matrix} F^{I} \\ M^{I} \\ F^{J} \\ M^{J} \end{matrix}}_{i} = {\begin{matrix} - Q_{x} (x_{i} = 0) \\ M_{xx} (x_{i} = 0) \\ Q_{x} (x_{i} = L_{i}) \\ - M_{xx} (x_{i} = L_{i}) \end{matrix}}_{i} = \hat{D} {\begin{matrix} \frac{\partial^{3} w_{0}}{\partial x^{3}} \\ - \frac{\partial^{2} w_{0}}{\partial x^{2}} \\ - \frac{\partial^{3} w_{0}}{\partial x^{3}} \\ \frac{\partial^{2} w_{0}}{\partial x^{2}} \end{matrix}}_{i} \end{matrix}

(9)

And using equation (8)

\begin{matrix} \underset{{F}}{\underset{︸}{{\begin{matrix} F^{I} \\ M^{I} \\ F^{J} \\ M^{J} \end{matrix}}_{i}}} = \underset{[B]}{\underset{︸}{{\hat{D}}_{i} [\begin{matrix} 6 & 0 & 0 & 0 \\ 0 & - 2 & 0 & 0 \\ - 6 & 0 & 0 & 0 \\ 6 L_{i} & 2 L_{i} & 0 & 0 \end{matrix}]}} \underset{{c}}{\underset{︸}{{\begin{matrix} C_{1}^{i} \\ C_{2}^{i} \\ C_{3}^{i} \\ C_{4}^{i} \end{matrix}}}} + \underset{{q}}{\underset{︸}{p_{i} {\begin{matrix} 0 \\ 0 \\ - L_{i} \\ \frac{L_{i}^{2}}{2} \end{matrix}}}} \end{matrix}

(10)

It is also possible to relate the nodal DoFs to the $C^{i}$ constants. Noting that $θ_{0} (x) = d w_{0} / dx$

\begin{matrix} \underset{{p}}{\underset{︸}{{\begin{matrix} w_{0}^{I} \\ θ_{0}^{I} \\ w_{0}^{J} \\ θ_{0}^{J} \end{matrix}}_{i}}} = {\begin{matrix} w_{0} (x_{i} = 0) \\ w_{0'} (x_{i} = 0) \\ w_{0} (x_{i} = L_{i}) \\ w_{0'} (x_{i} = L_{i}) \end{matrix}}_{i} = \\ \underset{[C]}{\underset{︸}{[\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ L_{i}^{3} & L_{i}^{2} & L_{i} & 1 \\ 3 L_{i}^{2} & 2 L_{i} & 1 & 0 \end{matrix}]}} {\begin{matrix} C_{1}^{i} \\ C_{2}^{i} \\ C_{3}^{i} \\ C_{4}^{i} \end{matrix}} + p_{i} \underset{{g}}{\underset{︸}{{\begin{matrix} 0 \\ 0 \\ \frac{L_{i}^{4}}{24 {\hat{D}}_{i}} \\ \frac{L_{i}^{3}}{6 {\hat{D}}_{i}} \end{matrix}}}} . \end{matrix}

(11)

Equations (10) and (11) can be written in compact form as follows

{F} = [B] {c} + {q}

(12)

{p} = [C] {c} + {g}

(13)

where ${F}$ is the vector of nodal forces, ${p}$ is the vector of nodal DoFs and ${c}$ is the vector containing the $C^{i}$ analytic constants. Solving equation (13) for ${c}$ and introducing into equation (12) yields a familiar FEM-like representation

[K] {p} = {F} + {f}

(14)

where $[K]$ is the stiffness matrix and ${f}$ is the vector of distributed loads. Again, there is a resemblance between the AEM equation for a plate strip element under cylindrical bending and the FEM equation of a classical beam element. The relation with the original matrices is given by

[K] = [B] [C]^{- 1} {p}

(15)

{f} = ([B] [C]^{- 1} - [I]) {g}

(16)

where $[I]$ is the $4 \times 4$ identity matrix. Given the parallelism between the AEM and FEM approaches, the AEM solver implementation is indeed akin to that of the FEM for the algorithms of assembly and solving of the global system of equations. However, once the global system of equations has been solved and the nodal DoFs have been elementwise reconstructed, finding the deflection ${w_{0}}_{i}$ requires computing the analytic constants by solving equation (13) for ${c}$ . On the other hand, the FEM requires a linear combination of shape functions using each element’s nodal DoFs as scalar coefficients.

3.4. Optimization module

The optimization module uses the Nelder–Mead simplex direct search method by Lagarias et al. (1998) and includes variable transformation to address bound constraints for the optimization parameters. This can be used to define both design criteria constraints (e.g. minimum active layer thickness and chamber length) as well as constraints imposed by pressure limits inside each chamber. The design parameters include

$α_{i}$ – the relative position of each chamber interface as a percentage of the total spoiler chord;

$h_{i}$ – the thickness of the upper skin (active layer) of each chamber;

$k_{i}$ – the spring constant modelling the stiffness of the chamber interface;

$p_{i}$ – the actuation pressure in each chamber.

Figure 12 shows how the structural and optimization modules are integrated into the design tool algorithm.

The base and target shapes are first loaded and the user inputs the number of pressurized chambers. Each chamber is associated with a series of new optimization parameters. Adding chambers has, therefore, a very significant impact on the problem complexity. Moreover, increasing the number of optimization parameters has a disproportional effect on the number of calculations the optimization algorithm must perform and consequently leads to a rapid growth of the algorithm runtime.

The value $f^{(j)}$ refers to the objective function value at the end of the jth iteration.

3.5. Objective functions

There are multiple ways of describing the conformity status between the candidate and target shapes. The sensitivity of each possible implementation of the objective function to the design variables will have a profound effect on the algorithm efficiency and accuracy. Moreover, since the simplex method does not ultimately converge to a minimizer for general (nonconvex) functions, a poor choice of objective function might even prevent reaching a solution. For this reason, the performance of different objective functions has been evaluated.

3.5.1. Maximum distance

A simple way to evaluate geometrical conformity is to use an objective function that retrieves the maximum distance between the two curves describing the candidate and target bump shapes. Assuming a piecewise linear approximation of the target shape, the distance between a point $C_{m} = (x_{0}, z_{0})$ on the candidate shape to two consecutive points $T_{n} = (x_{n}, z_{n})$ and $T_{n + 1} = (x_{n + 1}, z_{n + 1})$ on the piecewise linear approximation of the target shape is

\begin{matrix} ∥ C_{m} \to T_{n}, T_{n + 1} ∥ = \\ \frac{| (z_{n + 1} - z_{n}) x_{0} - (x_{n + 1} - x_{n}) z_{0} + x_{n + 1} z_{n} + x_{n} z_{n + 1} |}{\sqrt{{(z_{n + 1} - z_{n})}^{2} + {(x_{n + 1} - x_{n})}^{2}}} \end{matrix}

(17)

If a line passing through $C_{m}$ and normal to the direction formed by points $T_{n}$ and $T_{n + 1}$ intersects the segment $\bar{T_{n} T_{n + 1}}$ , then equation (17) computes the minimum distance of point $C_{m}$ to the piecewise linear approximation of the target shape. Repeating this process for a set of M points along the candidate shape, one obtains a set of distances from points on the candidate shape to a map of those points on the target shape. The optimization problem is now minimizing the maximum distance in that set. The objective function is

\begin{matrix} f_{1} = max_{C_{m} \in Γ_{C}} {∥ C_{m} \to T_{n}, T_{n + 1} ∥}, & n = 1, 2, \dots, N \\ m = 1, 2, \dots, M \end{matrix}

(18)

Figure 13 shows how point $C_{m}$ can be mapped to line segment $\bar{T_{n} T_{n + 1}}$ but cannot be mapped to segments $\bar{T_{n - 1} T_{n}}$ and $\bar{T_{n + 1} T_{n + 2}}$ : the intersection point of the normal to those segments containing $C_{m}$ lies on an extension and not on the actual segments. Using the same reasoning, point $C_{m + 1}$ must be mapped to the segment $\bar{T_{n + 1} T_{n + 2}}$ .

3.5.2. Area difference

Another approach to evaluate geometrical conformity is to consider the area enclosed between the current and the target shapes. The optimization problem is now the minimization of that area. The objective function is

f_{2} = \int_{0}^{l_{B}} | f_{C}^{(j)} (x) - f_{B} (x) | d x

(19)

where $f_{C}^{(j)}$ is the shape function of the candidate shape at the end of the jth iteration and $f_{B}$ is the shape function of the target shape. Unlike the previous implementation, the conformity status at each iteration now contains information from the entire domain of the current shape versus only one point that might change location arbitrarily from one iteration to the next as only the information of a single point is used to determine the conformity status of the entire domain. Therefore, it is expected that $f_{2}$ might be able to increase the robustness and accelerate the optimization process.

As seen in Figure 14, the entire domain is used to compute $f^{(j)}$ . Assuming the chambers pressurization as the design variables, if iteration j leads to the scenario shown, a larger portion of the candidate shape is above the target shape and, as a result, the pressure values tested on the simplex $Δ_{j + 1}$ will lead to a reduction in pressure near the trailing edge and an increase in the leading edge, denoted by $p^{(j + 1)} ↓$ and $p^{(j + 1)} ↑$ , respectively. The absolute value must be used in equation (19) because the optimization problem deals with the minimization of f.

3.5.3. Weighted area difference

A variation on the last objective function can be made using a weight function $ϕ (x)$ . As mentioned, the correct position of the SCB crest has the strongest influence on its drag reduction potential. Unfortunately, the previous implementations are unable to enforce special priority to geometrical conformity around the crest position. Using a weight function $ϕ (x)$ that introduces a penalty by amplifying the area difference around the crest position, this is possible. The objective function is now

\begin{matrix} f_{3} = I (ϕ) = \int_{0}^{l_{B}} ϕ (x) | f_{C}^{(j)} (x) - f_{B} (x) | dx \\ \forall {ϕ | ϕ (x) \geq 0, x \in [0, l_{B}]} \end{matrix}

(20)

This implementation is particularly useful if the desired target shape is not a physical solution. In this case, the best physical approximation to the target shape might lead to a large offset of the crest position as shown in Figure 15.

As referred, a penalty shall be introduced in the objective function. This will be achieved through amplification of the area enclosed between the target and candidate shapes. As a result, the weight function $ϕ (x)$ should have a bell-like distribution around the chordwise crest position as shown in Figure 15. Moreover, the extent of the chordwise region which is prioritized should also be adjustable. Finally, the area difference to be prioritized shall be amplified by a factor $1 + β$ , while the area difference in the remaining domain will be normalized, that is, $β = 0$ . With these considerations, the weight function is given by

ϕ (x) = ϕ_{0} + ψ (x)

(21)

where $ϕ_{0}$ and $ψ (x)$ have be defined as

\begin{matrix} ϕ_{0} = 1 - β \frac{δ}{f (μ | μ, σ^{2}) - δ} \\ δ = min (f (x | μ, σ^{2}) : x \in [0, l_{B}]) \end{matrix}

(22)

ψ (x) = β \frac{f (x | μ, σ^{2})}{f (μ | μ, σ^{2}) - δ'}, β > 0

(23)

and $f (x | μ, σ^{2})$ is the probability density of the normal distribution

f (x | μ, σ^{2}) = \frac{1}{\sqrt{2 π σ^{2}}} e^{- \frac{{(x - μ)}^{2}}{2 σ^{2}}}

(24)

The normal distribution was chosen as it provides a convenient way to adapt the position of the weight function maximum using the mean $μ = c_{B}$ . Moreover, the extent of the prioritized area can be changed using the standard deviation $σ$ , given in terms of the bump length. equation (22) ensures that outside the prioritized domain $ϕ (x) \to 1$ , while equation (23) sets the maximum of $ϕ (x)$ to $1 + β$ .

Figure 16 illustrates the implementation of the weight function where the bump parametrizations are represented with solid lines with the corresponding weight functions $ϕ$ shown on top using dashed lines. In this case, $ϕ$ was computed using $σ = 0.05 l_{B}$ and $β = 2$ .

3.6. Objective function comparison

Since aerodynamic investigations usually use parametrization techniques without a physical background, this section studies the effect of using non-physical solutions of the structural model as target shapes on the geometrical conformity achievable using the proposed design tool. The different design objectives will lead to distinct results in all three implementations of the objective function. However, because the overall geometrical conformity of the functional layer to the prescribed target shape is actually not the critical design driver from a drag reduction perspective, the crest positioning attained by each objective function will also be discussed and used as selection criteria.

Figure 17 shows a comparison between target shape C and the physical solution used to generate target shape A containing the same crest position. The initial morph for the optimization process is also shown in black.

Figure 18 shows the evolution of the actuation pressure for a two chamber design. The optimization history reveals that using objective function $f_{1}$ leads to a slow convergence of the optimization process. Indeed, the design tool is about four times faster when using $f_{2}$ and $f_{3}$ . The difference in convergence rate between $f_{2}$ and $f_{3}$ is negligible with $f_{3}$ only marginally slower that $f_{2}$ .

Figure 19 shows the final morphs obtained using the three design objectives. The results are shown separately for a clearer visualization. The geometrical conformity using objective functions $f_{2}$ and $f_{3}$ is considerably better than that reached with $f_{1}$ . In case of the latter, the crest positioning and the deformation near the spoiler trailing edge are particularly poor. On the other hand, the overall geometrical conformity obtained with $f_{2}$ and $f_{3}$ is good. However, there is a perceptible difference in accuracy of the crest positioning between the two objective functions.

To better evaluate this difference, Figure 20 shows a detailed view of the crest region using the different design objectives. This time, the results are combined for a better comparison of the accuracy in crest positioning.

The optimization based on $f_{2}$ is able to improve both the chordwise positioning of the bump crest and the bump height compared to the optimization using $f_{1}$ . However, as hypothesized, the weighted optimization using $f_{3}$ offers the best accuracy in terms of crest positioning, both in terms of the chordwise crest position and bump height. For the present scenario, an amplification factor of $β = 6$ was used and $σ = 0.05 l_{B}$ . A further improvement could be achieved by increasing $β$ .

Table 2 summarizes the relative error in the approximation of the SCB crest in terms of crest length $δ c_{B}$ and bump height $δ h_{B}$ . The last column shows the $l^{2}$ -norm of the error as a measure of the overall crest deviation.

Table 2.

Crest adaptation results.

Objectivefunction	$δ c_{B}$ (%)	$δ h_{B}$ (%)	$δ_{t} = \sqrt{δ {c_{b}}^{2} + δ {h_{B}}^{2}}$ (%)
$f_{1}$	5.6330	−4.1551	6.9997
$f_{2}$	0.1266	−2.4351	2.4383
$f_{3}$	−0.646	−0.4621	0.7939

This analysis shows that not only do $f_{2}$ and $f_{3}$ reduce the optimization runtime but they also allow for better overall geometrical conformity and a more accurate crest positioning. Moreover, without a significant penalty in terms of optimization runtime, $f_{3}$ permits a greater control of the crest positioning compared to $f_{2}$ which is extremely relevant for the aerodynamic efficiency of the device.

4. 2D design

The adaptability characteristics of the proposed spoiler concept are now analysed. An uncurved 5 mm thick functional skin made of glass fibre reinforced polymer (GFRP) has been selected as the baseline geometry and the span is 10 m. The actuation mechanism is based on a two chamber spoiler design in order to determine if a compromise between design complexity and adaptability qualities can be struck without resorting to additional chambers.

The position and stiffness of the chamber interface play an important role on the achievable geometrical conformity. However, they are not actuation variables but variables of the structural design which must be fixed once the spoiler structure is derived.

As a feasibility demonstration, the adaptability analysis is based on a worst-case scenario: as the crest position moves towards the leading and trailing edges, the required stiffness of the chamber interface increases abruptly. However, for symmetrical bumps, it is null. Therefore, setting the design point of the spoiler to $(c_{B})_{design} = l_{B} / 2$ is specially challenging in terms of extended adaptability over a large range of crest lengths. This also requires setting the position of the chamber interface to $(α_{1})_{design} = 0.5$ . The stiffness of the chamber interface can be determined by defining an actuation limit $(c_{B})_{\lim .}$ and then, using the design tool, compute $(k_{1})_{design}$ that, subject to $(α_{1})_{design} = 0.5$ , offers the best geometrical conformity for the actuation limit $c_{B_{\lim .}}$ . In keeping with the worst-case scenario approach, a narrow actuation limit of $c_{B_{\lim .}} = 0.425 m$ was chosen which leads to $(k_{1})_{design}$ = $k_{1} (c_{B} = 0.425) |_{α = 0.5} \approx 1.7 \times 10^{5} N / m$ .

4.1. Deformation envelope

Figure 21 shows a series of target shapes (in red) containing a set of crest lengths $c_{B}$ . The effective range of crest lengths is actually double, since results on the rear portion of the spoiler are analogous and have been omitted for conciseness. The design tool was used to determine the best morph of the active layer to each of the target shapes using the actuation pressures $(p_{1}, p_{2})$ subject to $(α_{1})_{design} = 0.5$ and $(k_{1})_{design} \approx 1.7 \times 10^{5} N / m$ . The set of optimized shapes (in black) forms a deformation envelope which represents the manifold of educible morphs. The actuation limits are revealed when an increased chamber pressurization is unable to move the crest position closer to the leading edge, but instead causes the bump height to increase.

4.2. Actuation characteristics

The relation between the deformation envelope and the actuation pressure is shown in Figure 22. Since the actuation mechanism is driven by the chamber pressures, the envelope of these variables will be referred to as the actuation envelope.

The actuation envelope shows that the minimum actuation pressure occurs for $c_{B} = 0.5 m$ when the left ( $p_{1}$ ) and right ( $p_{2}$ ) chambers are equally pressurized. As the bumps become more asymmetrical, so does the chamber pressurization. This goes on until the right chamber is unpressurised and the actuation limits are reached. This also illustrates how the range of adaptability is limited by the design choices: if adaptability were the design driver, a lower $(c_{B})_{\lim .}$ would lead to a stiffer chamber interface, that is, higher $(k_{1})_{design}$ and an increased minimum actuation pressure. The result would be more leeway for asymmetrical pressurization, and therefore, wider actuation limits.

Figure 23 shows the envelope of the design objective ( $f_{3}$ ) which is a measure of geometrical deviation between target and optimized shapes. The black line shows the initial design space before fixing $α_{1}$ and $k_{1}$ . The blue line shows the design space after fixing $(α_{1})_{design}$ and, as expected, the geometrical conformity at $(c_{B})_{design} = 0.5$ m is unaffected. However, the design space is reduced and the geometrical conformity for other crest lengths is affected. The final actuation domain is represented by the red line: the geometrical conformity at the design point is now impacted as a drawback for the adaptability range imposed by setting the actuation limits $c_{B_{\lim .}}$ .

The feasibility study shows that, even for a challenging design scenario, it is possible to achieve an actuation domain of feasible crest lengths that corresponds to over 20% the SCB length.

5. 3D design

The cylindrical bending approximation, even though useful, does not capture the complete 3D behaviour of the spoiler. Consequently, important 3D effects are also analysed.

5.1. Spoiler tip design

The SCB actuation was reduced to a 2D problem assuming that the active layer could be modelled as a semi-infinite plate strip. This assumption implied that the spoiler tips were sufficiently separated so that the corresponding boundary conditions did not significantly influence the bending problem. In this section, the effect of different tip designs will be analysed considering two distinct configurations: a spoiler with an exterior casing (Figure 24(a)) and a spoiler with open ends (and thus free tips) (Figure 24(b)).

Figure 25 shows the evolution of the deflection of the half-span crest located at $(l_{B} / 2, s_{sp} / 2)$ and 90%-span mean deflection. The limits on the mean deflection ${\bar{w}}_{0}^{α}$ for the two configurations are shown as shaded areas denoted by ${\bar{w}}_{0}^{100 %}$ .

The effect of SCB AR on the bending stiffness of configurations (A) and (B) is illustrated by the opposite evolution of the respective half-span crest deflections (in black) for lower ARs. For AR > 4, the half-span crest reaches the cylindrical bending prediction and both configurations exhibit similar deflections. Although the half-span crests of both configurations eventually reach the same deflection, which is not an indication that they are then aerodynamically equivalent.

Figure 26 illustrates the relative difference between the deflection of the half-span crest and 90% and 100%-span mean deflections. Indeed, both configurations exhibit a maximum for SCBs close to unit AR. The difference continuously decreases with increasing AR. Nonetheless, the configuration with a bonded casing converges much slower to a spanwise uniform bump, hence becoming less aerodynamically efficient than configuration (B) for a given AR. Moreover, 90% of the spoiler span of the configuration with free tips can be considered to uniformly deploy the functional layer at around AR = 2 ( $δ < 1 %$ ) while a similar efficiency using a bonded casing requires SCBs of ARs between eight and nine. Another relevant limitation of using a bonded casing is the fact that its application to inboard spoilers – which typically present ARs around two or three – can significantly lower the SCB drag reduction potential (between 10% and 20%) when compared to configuration (B). Finally, Figures 25 and 26 show that even for high AR SCBs – where the two configurations perform similarly across 90% of the spoiler span – using a bonded casing inevitably implies a loss of at least 5% in drag reduction potential due to the external 10% of the spoiler span.

5.2. Tapered SCB

Tapering is another 3D effect that has relevant implications on the actuation mechanism. Tapering ratios as low as 0.5 and 0.8 are under investigation in the LDAinOp program for the outboard and inboard spoilers, respectively. Figure 27 shows the crest deflection for different taper ratios. The spoiler was modelled with free tips.

For moderate taper ratios ( $λ_{sp} \geq 0.8$ ) and independently of AR, the spanwise deflection of the crest reduces linearly from the spoiler root ( $\tilde{y} = 0$ ) to the tip ( $\tilde{y} = 1$ ).

Figure 28 shows the effect of tapering on the mean crest deflections. Although there is a significant change in deflection of the half-span crest due to tapering, for a given taper ratio, the mean deflection across the entire spoiler span does not deviate significantly.

Given this result, it is reasonable to consider that even though the actuation pressure must be corrected in order to account for tapering, this might be achieved by scaling it proportionately to ${\bar{w}}_{0}^{100 %}$ and uniformly across the spoiler span. However, Figure 28 also shows that, even for low taper ratios, ${\bar{w}}_{0}^{90 %}$ deviates considerably from the half-span crest deflection. This means that the functional layer might be on average at a deflection close to that of the half-span crest, but there are however large deflection gradients across its span. Therefore, a uniform correction of the actuation pressure would lead to considerably lower deflections of the spoiler region close to the tips. Conversely, the region closer to the spoiler root would experience significantly larger deflections. While the former situation translates to a lower drag reduction potential, the latter has more serious consequences: overestimating the bump height not only leads to an accentuated loss of drag reduction potential but it might also lead to added drag when compared to a clean aerofoil, which would defeat the purpose of having an SCB altogether.

As a consequence, using tapering requires not only correcting the actuation pressure but also as demonstrated, this correction must be done using at least two to three sections of the spoiler span with progressively higher pressurizations.

5.3. Uniformization of tip deformation

The behaviour of the non-uniform spanwise deformation near the tip region of the active layer on a spoiler with free tips can be adjusted using only the offset distance introduced in Figure 7. The portion of the tips enclosed by $δ_{tip}$ is unpressurised

Adjusting the offset distance allows the tip deflection to be changed considerably without affecting either the half-span crest or the mean deflection.

Figure 29 shows the effect of the offset distance on tip deformation. The deflection of the half-span crest is depicted in red as a reference.

Recalling that an overestimation of the bump height can lead not only to lower drag reduction potential but also in the worst-case scenario added drag, the offset distance is an important design variable that allows the aerodynamicist to control eventual localized drag penalties in the tip region without resorting to more complex tip designs such as a spoiler with a bonded casing.

6. Demonstrator and preliminary feasibility study

The development of an initial demonstrator for the proposed integral spoiler concept has already started. Figure 30 shows an early prototype for a single pressurized chamber as an independent part designed to be easily swappable.

A preliminary numerical investigation on the feasibility of this pneumatic concept was also carried for a section of the outboard spoiler in cruise condition and maximum bump height. Figure 31 shows a simplified FEM model of a complete spoiler segment spanning 1.5 m inwards from the section under investigation and a detailed view of the flexible GFRP stringer used as the chamber interface.

Woven carbon fibre reinforced polymer (CFRP) and unidirectional GFRP (UD-GFRP) were chosen for the support structure and active layer, respectively. The pressurization level selected in each chamber is such that, under cruise aerodynamic loads, a specified maximum bump height of 8 mm is reached.

The resulting deformation is shown in Figure 32, leading to a maximum total deformation of 8.16 mm and a maximum principal strain of 0.45%.

It should be noted that, since the design is at a low technology readiness level (TRL), a comprehensive weight assessment and optimization was not part of this feasibility study. At a later stage, in addition to the cruise condition, a detailed analysis of the deformation induced by the spoiler deployment should be carried as well as a study on the dependency of the flutter margin upon the chamber pressurization.

7. Concluding remarks

Given that the successful implementation of shock control with an SCB requires an adequate adaptation of the crest position due to the changing flight conditions, a concept with an actuation system based on differential pressurization of a functional surface was introduced.

A design methodology for a two chamber design was presented and the adaptability analysis for the challenging case of a symmetrical bump at the design point revealed that this simple actuation mechanism could still achieve a domain of realizable crest lengths of over 20% the SCB length.

The effect of the design objective on the structural optimization performed by the design tool developed was studied using different objective functions. The robustness and efficiency of this process were increased with the introduction of an area-based objective, which translated to a convergence about 5.5 times faster than a distance-based one. In addition, the use of the proposed weight function allowed a more accurate positioning of the SCB crest, which strongly influences its aerodynamic efficiency. Overall, the crest deviation was reduced 6% by the choice of a sensible objective.

3D effects were also studied. First, it was shown that the mean deflection of 90% of the spoiler span deviates less than 1% from the half-span crest deflection for SCBs with a free-tip design and near unit ARs, while a similar deviation for a spoiler with a bonded casing requires ARs greater than eight. The effect of the complex tip deformation of the remaining 10% of the span upon the mean deflection of the entire spoiler is not greater than 1% in case of the former tip design and not less than 5% for the latter.

Second, tapered SCBs were analysed. For taper ratios greater than 0.8, the spanwise reduction in crest deflection is approximately linear, setting the case for the use of multiple chambers along the span with increasing pressure levels as a means of correcting the average deflection of each section due to the increased bending stiffness towards the spoiler tips.

Finally, it was shown that the use of an offset distance for the chamber assembly of the free-tip spoiler allowed for an uniformization of tip deformation with a negligible effect on the mean deflection.

Regarding the limitations of this study, an initial numerical investigation on the effect of the aerodynamic and actuation loads on the idealized support structure was carried for the cruise condition. However, a detailed fatigue analysis must be conducted in the future for all flight phases to ensure operation compatible with the airframe lifetime. At a higher TRL, the development of a wing predesign tool is also necessary to estimate the additional weight required by stiffeners due to the torque box volume consumed, as well as an assessment of panel flutter and mitigation strategies in case of induced aileron buzz.

Nevertheless, the preliminary feasibility study suggests the hypothesis that a compromise between design complexity and adaptability qualities can indeed be struck for a two chamber configuration. This leads to a relatively simple actuation mechanism, especially when compared to previous concepts such as the ones proposed by Wadehn et al. (2002) and Bein et al. (2000) which require a large number of actuators and complex control systems.

The efficiency of the design tool here presented also makes it a good candidate for future integration in a fully coupled aerodynamic-structural optimization environment so that the structural module approximates aerodynamically optimized target shapes. This will allow a detailed trade-off study on the structural constraints related to the proposed actuation mechanism and the drag reduction potential of the SCB.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was a part of the LDAinOp program and was funded by the Fifth Federal Aeronautical Research Program (LUFO V) by the Federal Ministry of Economic Affairs and Energy (BMWi) in the project LDAinOp (FkZ: 20A1302B).

ORCID iD

Afzal Suleman

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