In this study, the auxiliary spar is positioned at 63% of the airfoil chord from the leading edge, as illustrated in Supplementary Figure S1 To generate the morphed section (Supplementary Figure S2), the flap chord is divided into 12 vertical lines (ribs). Here, the intersections of the ribs with the camber line are referred to as discretized camber line points (DCLP). The camber line is then approximated by straight lines connecting the DCLPs, with each straight line, forming an angle with its adjacent line. Considering the flap’s upper and lower surfaces and ribs as a four-bar mechanism, altering the angle of the discretized camber lines results in different flap geometries. Supplementary Figure S3 demonstrates the variation of camber lines obtained using this mechanism, while Supplementary Figure S4 represents the corresponding flap geometries.

The flap nomenclature in this study is based on the position of maximum camber (x/c') and its maximum value (t/c') relative to the flap chord (Supplementary Figure S5). As depicted in Supplementary Figure S6, our flap is named TnCy, where Tn represents t/c' = n × 100, and Cy denotes the curve number proportional to x/c' with values ranging from 1 to 5. Supplementary Table S1 provides the curve numbers and their corresponding x/c' values.

To create channels, the flap’s upper surface is cut at nodes 2 and 8, while the lower surface is cut at nodes w and 4 (Supplementary Figure S2). This division results in three parts: the main airfoil body and flap parts 1 and 2. The downward vertical movement of parts 1 and 2 generates inlets and outlets of the slots (Figure 1). The slots function as channels to transfer and accelerate air from the lower surface of the flap to the upper, thereby preventing separation. The adaptability of the inlet is presumed to play an important role in facilitating smooth airflow transfer, while it is expected that the outlet area remains parallel to the flap’s upper surface (to ensure airflow attachment). Consequently, the angles of the channel walls are adjustable. As depicted in Figure 1, designing the walls of the channels involves obtaining the optimum curves between the nodes (AB and CD in channel No. 1; EF and GH in channel No. 2). This aspect is of considerable importance. The airfoil comprises three parts: the airfoil body, flap parts 1 and 2, and the nearest channel to the main airfoil in channel No. 1. Note, here the area covering the slot which opens during deflection is called Kool (shoulder).

Supplementary Figure S7 illustrates three different flaps with the same maximum camber, but with different maximum camber positions. Note, when going from left to right in this figure, the maximum camber position moves towards the trailing edge.

The parameters of the morphed flap are as follows:

1. Size of the maximum camber (t/c'),

Grid generation was performed using POINTWISE software. An O-type domain with a radius of 25c was utilized. The airfoil cell size at the leading edge was set to be 0.0001c to capture high-gradient zones accurately. The cell size at the flaps’ leading and trailing edges was set to be 1.047e-6c. The height of the first cell was chosen to achieve a y + value of 0.8. The quality of the grid was assessed based on mesh orthogonality, skewness, aspect ratio, etc. The number of cells in the boundary layer, wake zone, and far field was optimized to maintain a consistent lift coefficient (optimal number of cells was 210,000). Figures 2, 3 show the details of the grid and its resolution study.

For code validation, numerical simulations were performed for the NLR7301 airfoil, which has been previously studied experimentally in reference (Vandenberg and Oskam, 1980). Figure 4A shows a comparison of the pressure coefficient (Cp) between the numerical solution of this work and the experimental data. The present lift and drag coefficients differ from the corresponding experimental data by less than about 5 percent. Figure 4B displays the velocity field around the NLR7301 airfoil. The velocity contour depicted in this figure exhibits identical characteristics to the numerical simulation conducted in reference (Narsipur et al., 2012).

Figure 5 compares the velocity contour of the morphed flap in two different configurations (with and without channels) at a 5° angle of attack. In the configuration without channels, a large separation zone is observed (even at low angles of attack). By deploying channels, a jet flow is formed, and the separation region disappears, resulting in a significant increase in lift. However, drag nearly remains the same in both configurations. Note, in the configuration without channels, drag is mainly caused by high pressure resulting from separation. With channels, separation is minimized, and thus the pressure drag is considerably reduced (but the channels themselves become the main sources of drag). These two sources of drag appear to have significant interactions, resulting in a relatively consistent total drag. Table 1 presents the aerodynamic characteristics of the two configurations. The pitching moment coefficient is calculated about the quarter-chord point.

The following tests were conducted to design the channels’ shapes at three different angles of attack: 0°, 5°, and 10°, to analyze the intake of channels. Notation “∡A” represents the angle of the channel wall with respect to the horizon at point “A” (e.g., the angle of point “H” in Figure 1). The following cases were studied:

case 1

intake angle suitable for AOA = 0°:

∡B = ∡D = ∡F = ∡H = 0°, case 2. Intake angle suitable for AOA = 5°:

∡B = ∡D = ∡F = ∡H = 5°, case 3. Intake angle suitable for AOA = 10°:

∡B = ∡D = ∡F = ∡H = 10°, case 4. angle of the lower wall is 10° and the upper wall is tangent to the airfoil surface:

∡D = ∡H = 10°.

∡B = ∡F = tangent to the airfoil surface, and

case 5

angle of the lower wall is 10° and angle of the upper wall is 5°:

∡D = ∡H = 10°.

∡B = ∡F = 5°.

Figure 6 displays the aerodynamic characteristics of these cases, in which case 3 demonstrates the best performance. (Figures 7A, C) depict the velocity field of case 1, where even at zero angle of attack, a separation region is present in the leading edge of flaps 1 and 2, resulting in reduced aerodynamic performance. As the inlet angle increases (cases 2 and 3), this separation region becomes smaller, leading to a reduction in drag. (Figures 7, D) present case 3 at 10° and 0° angles of attack, respectively. Increasing the inlet angle causes the channel throat to move towards the outlet, resulting in an increase in lift. Case 4 exhibits the worst performance due to the channel profile moving the throat towards the inlet. Apart from the aerodynamic advantages of the fixed inlet shape in case 3, it also reduces the related mechanism’s complexity, production cost, and the overall weight of the airfoil.

The channels’ profiles were studied considering the following cases:

1. Normal case, with d1 being 3 and d2 being 2 percent of the airfoil’s chord,

Figure 8 demonstrates that increasing the width of the channel leads to an increase in lift and a decrease in drag until an optimum point is reached. Further increasing the channel width reverses these effects. Additionally, increasing the channel width results in an increase in the moment coefficient, which does not reverse. Moving the throat towards the outlet increases lift, drag, and pitching moment, but reduces aerodynamic efficiency. The normal case exhibits the highest aerodynamic efficiency (lift-to-drag ratio), while case 6 shows the lowest.

(Figures 9A–C) illustrate the lift coefficient versus maximum camber position for three different maximum camber values and three different angles of attack (0°, 5°, and 10°). When the maximum camber position is moved towards the trailing edge, the lift coefficient initially increases and then decreases, with the best position being at 50 percent of the flap chord. Increasing the maximum camber at a fixed maximum camber position also results in an increase in Cl.

(Figures 9D–F) represent changes in drag coefficient with maximum camber position for three different maximum camber values and three different angles of attack (0°, 5°, and 10°). Moving the maximum camber position towards the trailing edge leads to an increase in Cd. At a fixed maximum camber position, increasing maximum camber causes Cd to increase (independent of the angle of attack). The maximum camber position at 16.67 percent exhibits a cove that creates a dead air area, resulting in a large Cd value. Increasing maximum camber further increases both the dead air area and Cd. The maximum camber position at 33.33 percent with a camber of 4 percent shows the minimum Cd, regardless of the angle of attack.

For x/c' = 66.67 percent, both Cl and Cd are lower than those for x/c' = 83.33 percent due to an effect called hook-like (Supplementary Figure S8). This effect occurs when the camber position moves to 83.33 percent of the flap chord, resulting in an aerodynamic behavior similar to a gurney flap of references (Papadakis et al., 1996; Papadakis et al., 1997), leading to higher Cl and Cd.

(Figures 10A–C) depict aerodynamic efficiency versus maximum camber position for three different maximum cambers and for three different angles of attack. The optimal lift-to-drag ratio is found to be independent of maximum camber position and angle of attack always occurring at 33.33 percent of the flap chord (curve No. 2; C2). Generally, increasing maximum camber reduces l/d and the difference in l/d for various camber values decreases at higher angles of attack.

(Figures 10D–F) demonstrates the pitching moment coefficient versus maximum camber position for three different maximum cambers and for three different angles of attack. C_m behaves similarly to C_l, showing a hook-like effect as well. Lower maximum camber values result in lower C_m. Moving the camber position towards the flap’s leading edge leads to a smaller C_m. The minimum C_m is associated with the maximum camber position closest to the flap’s leading edge.

Supplementary Tables S2, S3 provide the best and worst camber positions and camber values for various aerodynamic characteristics, respectively. The desired flap geometry for each aerodynamic specification is also listed.

Figure 11 presents a comparative analysis of four distinct desired flaps, demonstrating a reduction of approximately 5 percent in the l/d difference at higher angles of attack. It is important to note that the selection of flaps may vary depending on the aircraft and its flight type. For example, during take-off and landing configurations, quantities related to higher angles of attack hold significant importance. In such scenarios, emphasis is placed more on Cl rather than Cd, Cm, and l/d. Consequently, the T8C3 flap is considered suitable for shorter runways.

To provide a practical example, let’s consider an aircraft that prioritizes drag reduction. Despite the higher Cl value of the T8C3 flap, an aircraft with specifications M = 300tons, SW = 511 m², AR = 7, and e = 0.8 lead to a take-off speed that is 3.28 km/h higher compared to the T6C3 flap. However, the T8C3 flap results in 3 percent higher drag force. On the other hand, the T4C2 flap exhibits a take-off speed that is 5.4 km/h higher, but a drag force that is 4.5 percent lower than that of the T8C3 flap. Additionally, the T4C2 flap’s smaller Cm leads to a reduced size of the horizontal tail, resulting in less drag and a negative lift force.

Figure 12 showcases the Cp distribution of the T8C3 flap, where fluctuations in Cp are observed on the upper surface of the flaps. These fluctuations are likely a result of interactions between the separation zone and the jet flow formed by the channels, leading to a mixing layer flow.

Figure 13 provide insight into the velocity field of the T8C3 flap. The geometry exhibits a separation zone at the trailing edge of flap No. 2. As the angle of attack increases, the separation zone remains constant due to the presence of the jet flow formed by channel No. 2, which inhibits separation. This behavior persists until the angle of attack reaches a limit where flow separation occurs at the Kool area.

At 30° flap deflection, Figures 14A–C depict the relationship between maximum camber position and Cl for two different maximum cambers and three different angles of attack. At this deflection angle, the camber position that yields maximum Cl shifts towards the trailing edge, precisely at half the flap chord (x/c' = 50%). It should be noted that a maximum camber of 6 percent, compared to 8 percent at zero angle of attack, demonstrates superior performance. However, as the angle of attack increases, a reversing phenomenon occurs. The reason behind this is that the lower maximum camber (6 percent) causes the Kool area to have larger angles relative to the flow, resulting in enhanced flow separation. Moreover, a flap with 6 percent maximum camber at 7° angle of attack experiences a deep stall.

(Figures 14D–F) showcase Cd versus maximum camber position for two different maximum cambers and three different angles of attack. Similar to Cl, the reversing phenomenon is observable here as well. While maximum camber position movement has a minimal impact on drag value, it generally leads to an increase in Cd as it approaches the trailing edge. For angles of attack below stall, the optimal maximum camber position is at the flap mid chord.

Figure 15, presented in panels a) to c), illustrates the relationship between l/d and maximum camber position for two different maximum cambers at three distinct angles of attack. Similar to Cl and Cd, smaller maximum camber values exhibit better performance at lower angles of attack. Although the movement of the maximum camber position has a slight impact on l/d, generally, when it shifts towards the trailing edge, l/d decreases. In comparison to the 15° deflection, the position of maximum l/d has shifted from 33.33 to 50 percent of the flap chord.

(Figures 15D–F) depict the behavior of Cm in relation to maximum camber position. The behavior of Cm aligns with other aerodynamic characteristics. For instance, there is a small difference between the 6 percent and 8 percent maximum camber values, and a reversing phenomenon is observed as well.

Figure 16 provide a comparison of the aerodynamic performance of flaps with varying maximum camber position. It is evident that the flap maximum camber position located in the middle of the flap chord yields the best performance. Hence, for the 30° morphed flap deflection, the desired configuration is the T8C3 flap.

Figure 17 displays Cp distribution over the airfoil surface for three different angles of attack, highlighting the fluctuations resulting from the mixing layer flow on the upper surface of the flap.

Figure 18 showcase the flow field, even at zero angle of attack, indicating a separation zone at the trailing edge of flap “2”due to the steep slope of the upper surface in that area. Additionally, a larger dead region is observed at the leading edge of flap “1”caused by the alteration in the channel path, where the separated flow needs to travel a longer distance to reach the channel wall. In all angles of attack, the presence of the jet flow generated by channel number 2 maintains the trailing edge separation unchanged. At 7° angle of attack, corresponding to the stall angle of attack, separation initiates at the flap Kool area.

(Figure 19A) demonstrates that even at zero angle of attack, flow separation occurs for the T4C3 flap. To delay separation in the Kool zone, modifications were implemented. Reducing the angle of the Kool zone increases the outlet width of channel “1,” and thus, a third part was added to maintain the outlet width unchanged. Furthermore, to reduce the length of the Kool zone, the outlet of channel “1” was shifted from node number 2 to node number 1. (Figure 19B) visually presents the related changes, and Table 2 provides the results of the geometry modifications.

Figure 20 compares the aerodynamic characteristics of the T8C3 and modified T4C3 flaps. Although the T8C3 flap outperforms the T4C3 flap in terms of aerodynamic performance, it is worth considering that the T8C3 flap has an 8 percent maximum camber, resulting in a larger Cl. However, when considering Cl, the T4C3 flap exhibits a better improvement compared to the T8C3 flap, along with an increased stall angle of attack.

Figure 21 showcase the velocity field of the modified T4C3 flap, clearly illustrating the disappearance of separation. It is important to note that the addition of an extra part to the morphed flap introduces challenges related to increased weight, cost, and production complexities.