Energy considerations and flow fields over whiffling-inspired wings

Some bird species fly inverted, or whiffle, to lose altitude. Inverted flight twists the primary flight feathers, creating gaps along the wing’s trailing edge and decreasing lift. It is speculated that feather rotation-inspired gaps could be used as control surfaces on uncrewed aerial vehicles (UAVs). When implemented on one semi-span of a UAV wing, the gaps produce roll due to the asymmetric lift distribution. However, the understanding of the fluid mechanics and actuation requirements of this novel gapped wing were rudimentary. Here, we use a commercial computational fluid dynamics solver to model a gapped wing, compare its analytically estimated work requirements to an aileron, and identify the impacts of key aerodynamic mechanisms. An experimental validation shows that the results agree well with previous findings. We also find that the gaps re-energize the boundary layer over the suction side of the trailing edge, delaying stall of the gapped wing. Further, the gaps produce vortices distributed along the wingspan. This vortex behavior creates a beneficial lift distribution that produces comparable roll and less yaw than the aileron. The gap vortices also inform the change in the control surface’s roll effectiveness across angle of attack. Finally, the flow within a gap recirculates and creates negative pressure coefficients on the majority of the gap face. The result is a suction force on the gap face that increases with angle of attack and requires work to hold the gaps open. Overall, the gapped wing requires higher actuation work than the aileron at low rolling moment coefficients. However, above rolling moment coefficients of 0.0182, the gapped wing requires less work and ultimately produces a higher maximum rolling moment coefficient. Despite the variable control effectiveness, the data suggest that the gapped wing could be a useful roll control surface for energy-constrained UAVs at high lift coefficients.


Nomenclature
Acover area of the vertical gap cover face Agap area of the full vertical gap face bg width of one gap F total force on all nine vertical gap faces in the +y direction H aileron hinge moment N cells number of cells in the volume mesh R ratio of gap cover area to total gap face area W aileron work to actuate the aileron Wgap covers work to actuate gaps, based on gap cover area δa aileron deflection angle

Introduction
Some species of birds, such as waterfowl, invert midflight while keeping their heads level in a maneuver known as whiffling, or 'dumping' [1][2][3]. Birds have been observed whiffling to evade predators and descend rapidly, among other reasons [1][2][3]. Whiffling has been qualitatively described as 'a swift zigzagging, side-slipping erratic fall' [1], 'aerobatic' [3], and losing altitude 'at a rapid rate' [3]. Amateur videos of whiffling show similar qualities [4]. During whiffling, the direction of airflow from the back to the belly side of the bird causes the primary flight feathers to twist open [5]. The resulting gaps between the feathers likely allow air to pass through the wing, leading to decreased lift and descent (figure 1). While whiffling has not been rigorously studied in the biological or engineering domains, it has been used as an engineering example of head-stabilization [6]. However, the maneuver has yet to inspire aircraft design. Due to its high maneuverability and rapid descent, whiffling could be an advantageous new source of bio-inspiration for aircraft control surfaces. The effects of whiffling on a flight path can be compared to two control surfaces found on uncrewed aerial vehicles (UAVs) [3]: spoilers and ailerons. Like spoilers, whiffling decreases the lift-to-drag ratio and leads to descent. Whiffling just one wing, while not known to be possible for birds, could lead to a rolling moment like an aileron due to the resulting asymmetric lift distribution across the full wingspan. Whiffling-inspired control surfaces could also be beneficial in terms of energy costs. UAVs are energy-constrained [7,8], and birds are frequently either power-or energy-constrained [9]. Limited battery capacities restrict important UAV mission parameters like flight duration, range, and payload weight [7,8]. Thus, while maneuvering costs may represent a small portion of a UAV's total energy budget, it is nevertheless important to consider and minimize the energy consumption of all aspects of flight in order to make the most of battery capacity. Maneuvering costs would be particularly relevant for missions involving loitering or extensive actuation. Evaluating and reducing the energy cost of actuators for maneuvering is an active area of study [10,11]. The passive nature of feather rotation [5] and apparent ease of the whiffling maneuver suggest that a whiffling-inspired control surface may require less work than conventional deflecting control surfaces.
Inspired by these similarities and the possibility of energy savings, a previous experimental study investigated novel wings with different numbers of gaps in the trailing edge [12]. Whereas ailerons and spoilers deflected into the freestream, the gapped wings remained in the plane of the wing when actuated. The study found that the gapped wings were not a suitable alternative to spoilers [13], since they did not decrease lift as much as a spoiler and required greater work to actuate [12]. Conversely the results implied that the gapped wings could be beneficial for roll control at high lift coefficients, compared to a wing with a single deflected aileron [14].
Here, we used Siemens STAR-CCM+-a commercial computational fluid dynamics (CFD) software-to simulate the gapped wing and representative aileron. Due to differences in geometry and flow parameters between the gapped wing and aileron wing, we also simulated a third configuration that allowed us to directly compare the two control surfaces. The CFD simulations agreed well with the previous experimental data. The results highlight several important flow phenomena of the gapped wings. These mechanisms provided valuable insight into the aerodynamic performance, control effectiveness, and work requirements of the gapped wing. Further, we more accurately estimated the work required to actuate a gapped wing based on aerodynamic loading. The findings provided additional evidence that gapped wings may require less work than an aileron at high rolling moment coefficients, and may produce a higher maximum rolling moment coefficient. Note that this work focused on bioinspiration as opposed to biomimicry. While the gapped wings were inspired by whiffling, the results presented are not necessarily informative of the avian whiffling behavior or mechanisms.
This work presents significant findings that expand our understanding of the gapped wings. Due to the novel nature of the gapped wing, the overall goal was to provide an initial steady-flow CFD characterization of a gapped wing. To the best of our knowledge, this work presented the first computational study of flow through gaps along the chord of a wing, and an important step in assessing the capabilities of whiffling-inspired wings as control surfaces. Secondly, the simulations were critical for building confidence in the previous experimental results [12]. The CFD simulations also directly calculated aerodynamic forces on all surfaces of the wing, allowing us to estimate actuation work due to aerodynamic loading without costly prototypes, equipment, and instruments. This approach differed substantially from previous work estimates of the gapped wings. The simulations also enabled us to understand the fluid mechanics behind the performance, work requirements, and control effectiveness of the gapped wing. The experimental work identified the general performance of the gapped wings, and insight into how gaps affected the global forces and moments of the wing. The simulations then provided an understanding of underlying steady flow mechanisms of the gaps, without the need for complex experimental setups. Furthermore, simulating the gapped wing enabled future cost-effective and timely parameter sweeps, optimization studies, and design iterations.

Methods
Before conducting our CFD study, we defined the wing configurations to be simulated (table 1). The first configuration (SI-gapped wing) is the same wing geometry and flow parameters as Sigrest and Inman's previously conducted gapped wing experiments [12]. We also simulated a baseline SI wing without gaps (SI-baseline wing), which is discussed further in appendix A. The second configuration (JH-aileron wing) is the same wing geometry and flow parameters as Johnson and Hagerman's previously conducted aileron experiments [14]. These two configurations were used to experimentally validate the CFD simulations, and a mesh convergence study was conducted on the SI-gapped wing. However, the JH-aileron wing and SI-gapped wing had different geometries and flow parameters. To reconcile these differences, we simulated an aileron of the same relative dimensions as the JH-aileron, but with the wing geometry and flow conditions of the gapped wing (SI-aileron wing). Simulating the SI-aileron wing allowed us to directly compare rolling moment coefficients and actuation work of the gaps and aileron [14]. We also simulated gaps on a wing with the geometry and flow conditions of the JH-configuration (JH-gapped wing). The JH-gapped wing behaved very similarly to the SI-gapped wing. Therefore, we focused here on the SI-gapped wing. We addressed the JH-gapped wing at the ASME Conference on Smart Materials, Adaptive Structures, and Intelligent Systems [15]. While the previous experimental study compared the gapped wings to a spoiler and aileron [12], we focused solely on the aileron comparison because the gapped wings were not found to be beneficial for rapid descent. We also only simulated the wing with nine gaps, since it produced the greatest rolling moment coefficients in the previous study [12].
We used Siemens STAR-CCM+ to perform steady-state three-dimensional (3D) Reynoldsaveraged Navier-Stokes (RANS)-based CFD simulations. Compared to higher-fidelity CFD approaches such as scale-resolving simulations, RANS provided us with an appropriate balance between simulation accuracy and computational cost. Since this initial study sought to measure average force and moment values and capture the general effects of the gaps on the flow field, RANS was suitable for our needs. Future studies will perform more in-depth unsteady simulations of the gapped wing flow. We were primarily interested in high-lift configurations and chose the k-ω SST turbulence model for its ability to capture flow separation, particularly in the low-Re flow regime [16,17]. We used the uncoupled solver for the JH-aileron wing and SI-aileron wing, and the coupled solver for the SI-gapped wing since it presented more of a challenge to converge. We monitored the unscaled residuals and terminated the simulation after x-momentum, y-momentum, z-momentum, continuity, and turbulent kinetic energy (TKE) were all below 1 × 10 −4 , and after specific dissipation rate (SDR) was reduced by at least four orders of magnitude. In two cases, momentum and TKE only fell below 5.3 × 10 −4 , and in two other instances, SDR only relatively fell below 7.5 × 10 −4 . This typically took about 500 iterations for the gapped wing, between 200 and 850 iterations for the SI-aileron wing, and between 700 and 875 iterations for the JH-aileron wing.
The 3D volume mesh was generated using STAR-CCM+'s built-in tools (figure 2). We used an unstructured polyhedral mesh with extruded prism layers near the wing surface to capture the boundary layer. The prism layer parameters were chosen such that the wall y + value remained below one for the majority of the wing surface and the growth ratio was less than 1.3. The prism layers were also applied to the vertical gap faces. Wake refinement was applied about two chord lengths behind the wing and endplates.
Per standard practice, we calculated the nondimensional force and moment coefficients of the wings [19]. The reference areas and lengths (with units) that were used to calculate the dimensionless coefficients of each configuration are shown in table 1. These were also the values used to set up and run each simulation, as dimensional values were required by the software. Note that the reference area for the gapped wing was calculated as the planform area of the wing minus the planform area of the gaps. The endplates were considered thin and not included in the planform area. Moment coefficients were measured and reported about the quarter-chord at the wing root. The previously conducted experiments tested half-span models in a wind tunnel using the reflection plane methodology, then corrected the data to a full asymmetric wingspan with the control surface on only the right semi-span [12,14]. Conversely, we simulated the full asymmetric span in a bullet-shaped domain to avoid reflection plane corrections in post-processing. The CFD domain boundaries were 12.3 times the wingspan of the SI-gapped wing and SI-aileron wing and 10 times the wingspan of the JH-aileron wing. All data were reported in the wind axes [18]. According to sign conventions, a negative aileron deflection angle was upwards and a positive aileron hinge moment was downwards towards the neutral position [18,20]. Positive rolling moments were produced by an upwards aileron deflection and open gaps.

SI-gapped wing experimental validation
To experimentally validate the SI-gapped wing-the nine gap wing tested by Sigrest and Inman [12]we replicated the experimental geometry and flow conditions in CFD (table 1). Each gap had a length 2/3 of the wing chord and a width 1/48 of the halfspan. Additionally, we simulated the SI-baseline wing without gaps, presented in appendix A. The previous experimental work also provides an in-depth comparison of the behavior of the gapped wing and baseline wing [12]. In the previous experiment, two circular endplates were used to reduce tip effects and approximate two-dimensional flow over the wing. We included the endplates in the simulations to ensure accurate experimental validation. We simulated the SI-gapped wing at angles of attack from 0 • to 12 • . The simulated angle of attack range was limited to 12 • due to the tendency of RANS to under-predict stall angle.
The previous experiment and current CFD simulations used two different approaches to estimate the actuation work of the gapped wing. Both methods considered a model with 'gap covers' that slid along the span of the wing to cover the gaps (figure 3(A)) or open them ( figure 3(B)). In the previous experiment, the work to open the gaps was estimated as a friction force acting on the gap covers over the width of the gap, with the normal force estimated as the lift on the gap covers minus their weight [12]. In the current study, we sought a more accurate estimate based on aerodynamic loading as opposed to classical mechanics. We calculated the actuation work directly from aerodynamic forces on the gaps, combining the CFD results with an analytical approach. We assumed that only aerodynamic forces acted on the gap covers, and excluded gravity, friction, and other forces. From the CFD data, we measured the force acting on all nine vertical gap faces (light blue in figure 3(C)) in the spanwise direction. Then, we multiplied the force by the width of a gap. This quantity represented the work required to open the gaps by retracting the entire gap face into the wing, assuming a constant force over the actuation distance. We assumed a constant force because the models are static and rigid, thus the gaps remain a constant width. However, this work was calculated on a larger face than is realistic. The work should only be calculated on the area corresponding to the gap cover (dark brown in figure 3(C)), since only this gap cover would move during actuation. Therefore, we analytically estimated the work to actuate the gap covers by scaling by the ratio of the two areas R, according to: This equation represents the work required to slide just the gap covers to open the gaps. We analytically scaled the work because of the limitations of modelling small features in CFD. However, using equation (1) made the work estimate dependent on the geometry of the gapped wing model, specifically the area ratio R. Thus, the results were dependent on the assumed geometry of the gap covers. Here, we assumed an area ratio R of 0.0964 based on a CAD model with 0.794 mm (1/32 in.) thick gap covers.

JH-aileron wing experimental validation
We also experimentally validated the JH-aileron wing using the geometry and flow conditions from Johnson and Hagerman [14] (table 1). We assumed standard atmospheric conditions. We simulated a single aileron on the right semi-span, with a chord of 25% of wing chord and a span 24.2% of the semispan starting at 72.6% of the half span [14] because this is the same configuration used in [12] and these dimensions are within the typical range for an aileron [20]. We calculated the aileron hinge moment about the hinge axis at 75% of wing chord and normalized it by aileron area and aileron chord. Since the rolling moment coefficient of the aileron was largely constant across angle of attack [14], we simulated the wing at a constant 10 • angle of attack, varying aileron deflection angle from 0 • to −30 • . We estimated the work to deflect the aileron against aerodynamic loads by directly measuring the hinge moment from the CFD results, then multiplying it by deflection angle: Similarly to the gapped wing, this work estimate assumed that only aerodynamic forces acted on the aileron, and that the hinge moment was constant across deflection. The constant hinge moment assumption was due to the static and rigid nature of the model.

SI-aileron wing for direct comparison
There were several differences in the wing geometry and flow parameters of the JH-aileron wing and the SI-gapped wing (table 1). Thus, to directly compare the aileron with the gaps, we simulated an aileron on the right semi-span of a wing with the same geometry and flow conditions as Sigrest and Inman's wing (table 1). This SI-aileron wing had a 0.2286 m chord, 0.4064 m semi-span, and circular endplates [12]. The aileron retained the same relative dimensions as the JH-aileron wing: a width 24.2% of the semi-span and a length 25% of the chord with the hinge axis at 75% wing chord. We simulated this wing at angles of attack from 0 • to 10 • and with aileron deflections between 0 • and −30 • . We estimated the actuation work according to equation (3), assuming a constant hinge moment.
To keep our comparison between gapped wing and aileron wing actuation one-to-one, our estimates (equations (1) and (3)) did not include the work required to pitch either wing to the simulated angle of attack. We considered the gapped wing to already be positioned at the angle of attack when actuated. Thus, it was not necessary to include the pitch work in our estimates. Since the rolling moment coefficient of the aileron is insensitive to angle of attack [14], we were able to assume it was already angled at the moment of actuation and exclude the pitch work. In addition, angle of attack is typically controlled by an elevator, flaps, or other symmetrically deployed control surfaces. Estimating the work to change angle of attack would require modelling the full configuration of a hypothetical UAV. Making these assumptions to estimate pitch work would not necessarily improve the fidelity of the actuation work estimates, because such assumptions can greatly alter aircraft performance. Thus we simulated just the wings with the single relevant roll control surface, and neglected the work required to change the wing's angle of attack.
The work estimates for both the SI-aileron wing and SI-gapped wing were likely over-predictions because we assumed static and rigid models. We assumed a constant force over the gap cover displacement, and constant hinge moment over the aileron deflection. In reality, the hinge moment likely increases as the aileron deflects farther, and the force on the gap covers increase as they open. This is a topic for future investigations.

Mesh convergence study
We conducted a mesh refinement study on the SIgapped wing, for the lift coefficient and rolling moment coefficient at 3 • angle of attack (figure 4). Five meshes were generated with approximately 1.0 million, 2.5 million, 3.5 million, 6.1 million, and 9.0 million cells. The meshes were locally refined around areas of fine geometry, such as the gaps and endplate edges. We computed the grid factor of each mesh according to [21]: Neither the lift coefficient nor the rolling moment coefficient behave linearly as the mesh is refined, but this is expected due to uneven refinement of the polyhedral mesh and the complex flow field. In particular, the rolling moment coefficient results are in line with typical moment predictions in CFD. For example, the pitching moment coefficients computed as part of the AIAA CFD Drag Prediction Workshop show not only considerable spread between entries, but also differing trends [22]. Some solvers converge from below and others from above, and a substantial number of entries do not obtain a linear convergence region during refinement [22]. Based on the mesh refinement study, we use the 3.5 million cell mesh for the SIgapped wing, 2.6-3 million cells for the SI-aileron wing, and 2.2-2.8 million cells for the JH-aileron wing. The exact number of cells depends on aileron deflection and angle of attack. This refinement study ensured that we used an appropriate level of mesh density for each case. For the mesh density chosen, we found good agreement between the CFD simulations and previous experimental data.

Experimental validation
The SI-gapped wing CFD results generally agree well with the previous experimental results (figure 5) [12]. We simulated the wing at several angles of attack ranging from 0 • to 12 • . The lift coefficient predictions are within experimental uncertainty except for the data point at 10 • angle of attack, which differs from the experiment by 0.013, and the data point at 12 • angle of attack, which is partially converged likely indicating separated flow and stall. Excluding the stalled data point, the drag coefficient falls within experimental uncertainty. The rolling moment coefficient is generally within experimental uncertainty and differs by at most 0.0038. The 12 • stall angle of attack predicted by CFD is lower than the stall angle of 14.7 • angle of attack found experimentally. It is common for RANS to under-predict stall angle [23]. The experimental validation of the SI-baseline wing also matches well (appendix A).
The CFD results of the JH-aileron wing also agree well with the experimental data (figure 6) [14]. We compare simulated and experimental data across a range of upwards aileron deflections from 0 • to −30 • , at a constant 10 • angle of attack. The rolling moment coefficient differs from the experimental data by less than 0.0012, and the hinge moment coefficient differs by less than 0.022.

Aerodynamic effects of the gaps on the flow over the wing
First, we investigate the time-averaged aerodynamic effects of the gaps on the broader flow field over the wing. While there have been studies on the airflow over geometries related to the gapped wingincluding wings with serrated trailing edges [24,25], steps [26,27], and slotted airfoils [28,29]-this is the first instance of modelling flow over a wing with this gap geometry to the best of the authors' knowledge. The flow information presented could provide context and insight into steady flows over other similar geometries.
Compared to a baseline wing section, the presence of a gap affects the velocity and thickness of the local boundary layer, as measured at 10 • angle of attack (figure 7). The flow over the baseline wing section is nominal for a NACA 0012 airfoil. Conversely, the flow through the gap is more vertical, resulting in a thicker boundary layer. For instance, at 95% of the wing chord, the boundary layer is 0.010 m thick at Figure 5. The SI-gapped wing CFD results agree well with previous experimental data [12], considering the lift, rolling moment, and drag coefficient. The data point at 12 • angle of attack is partially converged, likely indicating stall. The gray transparent ribbon around the experimental data represents the uncertainty at an approximately 95% confidence level [12]. Figure 6. The JH-aileron wing CFD results agree well with previous experimental data [14], comparing the rolling moment and hinge moment coefficient at 10 • angle of attack. Negative deflection angles are upwards. the baseline section, and 0.026 m thick at the central gap root face section-2.5 times thicker than over the baseline section (appendix B). However, a much thinner portion of this boundary layer experiences very low velocities near the wing surface (dark blue  figure 7), compared to the boundary layer over the baseline section. The increased velocity within the boundary layer indicates that the flow through the gap re-energizes the boundary layer over the suction side of the trailing edge [29] (appendix B). This effect is similar to that seen with slotted airfoils, which inject energy into the boundary layer to delay separation and stall [28,29]. Therefore, the re-energization of the trailing edge flow could be one cause of the gapped wing's delayed stall, a performance benefit that was previously observed experimentally [12]. Note that our boundary layer analysis shows that the slower moving wake (blue shaded regions in figure 7) falls within the boundary layer (appendix B). The velocity magnitude outside of the boundary layer remains largely unaffected by the gap.
Each gap also produces a pair of trailing edge vortices, where the airflow is allowed to curl around each edge of the gap from the lower to the upper surface of the wing (figures 8 (A) and (B)). Each vortex pair produced by a single gap counter-rotates-the vortex nearest the wing root rotates counterclockwise and the vortex closer to the wingtip rotates clockwise (as viewed from the trailing edge). In figure 8, we quantify the strength of the vortices in terms of the Qcriterion, an established method of vortex detection based on the vorticity and strain rate tensors [30][31][32], as well as the magnitude of the vorticity. Higher Qcriterion and higher vorticity magnitude both indicate a stronger vortex, and vice versa. As the vortices travel downstream, the radius of the Q-criterion isosurfaces increases, and the vorticity magnitude decreases. Furthermore, the vorticity magnitude and size of the isosurfaces appear generally constant across the gaps, with one main exception: the isosurface nearest the wing root grows to a larger diameter and extends farther downstream. This inboard gap vortex is free to grow because it is far from large geometric features and flow structures. Conversely, the central gap vortices are bound by the endplate and most inboard gap vortex. Additionally, the mutual interactions between the central gap vortices may impact their size and dissipation. Generally, the vortices on the wing root side of the gaps also extend farther downstream than the vortices on the wingtip sides.
Importantly, the vortex behavior of the gapped wing clarifies why gapped wing control effectiveness varies over angle of attack (figure 5) [12]. The vortices overall decrease the lift produced by the gapped semi-span [19,20], thus causing roll due to the asymmetric lift distribution relative to the baseline semispan. The strength of the vortices increases at higher lift coefficients, which is proportional to the angle of attack for the gapped wings. The stronger vortices then lead to larger decreases in lift relative to the baseline wing, and higher rolling moment coefficients. In short, the relationship between angle of attack and vortex strength is closely connected to the variable roll control effectiveness of the gapped wing.
There are also substantial differences in the vortices produced by the gaps and the aileron (figure 8).
We investigate the vortices present on the SI-aileron wing deflected −15.3 • at 2 • angle of attack, as a comparison to the SI-gapped wing. Each aileron edge produces a vortex structure, with the two vortices counter-rotating. The vorticity magnitude of the aileron Q-criterion isosurfaces are comparable to that of the gap vortices. However, at both Q-criterion levels the aileron isosurfaces are larger in diameter than the gap vortices, even the largest gap vortex. In the case of the 1 s −1 Q-criterion, the aileron isosurfaces also extend further downstream than those of the central gaps. Furthermore, the vortical flow produced by the aileron is concentrated near the wingtip, while the gap vortices are distributed along the semispan with the largest vortex at the wing root.
The vortices produced by these two control surfaces inform their overall aerodynamic performance. Vortices affect the lift distribution of a wing. In turn, the distribution of induced drag is a function of lift distribution, and is responsible for the roll-yaw coupling traditionally seen with deflected ailerons [20]. A yawing moment in the opposite direction from roll is referred to as adverse yaw, since it results in an uncoordinated and inefficient turn [20]. Gapped wings were previously found to produce comparable rolling moment coefficients to an aileron, and ultimately a greater maximum rolling moment coefficient [12]. At the same time, the gapped wings produced negligible yaw compared to the aileron. These results imply that the gap vortices have a favorable impact on lift distribution-hence the distribution of induced drag-because the gapped wing produces comparable rolling moment coefficients to an aileron while minimizing drag. Ailerons are particularly susceptible to adverse yaw due to the strong vortices and asymmetric lift distribution far from the wing root [20]. However, the vortex arrangement of the gapped wings minimized this effect-producing less induced drag, especially farther outboard along the wingspan. Meanwhile, the vortices were not detrimental to roll production: the lift decrease from the gaps and their downwash was equivalent to the lift decrease caused by the aileron and its downwash.

Flow through a gap
With an understanding of how the gaps affect the broader flow field over a wing, we investigate the steady behavior of the flow through the gap itself. We examine the time-averaged flow through the central gap in the SI-gapped wing at 2 • and 10 • angle of attack. While we simulate the SI-gapped wing at a broader range of angles, we study these two cases more closely to understand the representative behavior at low and high angles of attack. We also examine the velocity field on two orthogonal planes, to enable us to characterize the streamwise and spanwise components of the flow individually. We measure the velocity and pressure in the central gap as a representative set, since it is far away from major geometrical features like the endplates or the transition to the baseline semi-span.
In general, the gapped wing decreases lift compared to the baseline wing by allowing airflow through the gaps from the pressure to the suction side. For completeness, appendix A experimentally validates the SI-baseline wing simulations, and briefly compares the pressure coefficients over the baseline wing to the gapped wing. The previous experimental work also thoroughly compares the gapped wing and baseline wing in terms of aerodynamic force and moment coefficients, descent, and roll [12]. Figure 9 illustrates the streamwise flow through the gap. The velocity field shows slow-moving recirculating flow at the start of the gap. Such recirculation regions are a general flow characteristic also seen in flows over a backwards-facing step [26,27]. As angle of attack increases from 2 • to 10 • , the flow through the gap becomes more vertical and the recirculation region appears to shorten in the freestream direction. This behavior is expected given that increasing angle of attack increases the vertical component of the freestream. Shortly downstream, the flow passes through the gap with an upward vertical component related to the angle of attack. As seen in figure 7, this re-energizes the boundary layer over the suction side of the trailing edge similarly to flow through a slot along the wing span [28,29]. Thus, the gaps may help delay the stall of the gapped wing. Figure 10 shows spanwise components of the flow in the gap (parts (A) and (C)). The flow within the gap itself appears to be largely in the streamwise direction, with noticeable spanwise components only at the recirculation region and trailing edge of the wing where the gap vortices form. The spanwise flow components may be small because of the narrow width of the gap; it is possible that spanwise flow structures do not have time to fully develop in the confined gap space. Future parameter sweeps on the gap geometry will help validate this hypothesis. The pressure contours on the gap face closest to the wing root also agree with the behavior of the velocity field (figure 10). The most negative pressure coefficients are at the start of the gap, corresponding to the recirculation region shown in figure 9. As the airflow moves aft along the chord, there is a slight favorable pressure gradient that reaches a minimum negative value towards the end of the recirculation region. Further downstream, the pressure gradient remains generally adverse, reaching first a local maximum at about 50%-60% chord at the end of the recirculation region, and then a global maximum pressure coefficient at the trailing edge. Overall, this pressure distribution shares a similar trend to that of a laminar separation bubble on an airfoil [33].
Notably, the fluid mechanics in the gap are related to the work requirements of the gap covers (equation (1), figure 10). The majority of the gap face experiences negative pressure coefficients, especially in the recirculation region. The pressures imply Note that the pressure distribution is relatively constant over the height of the gap in the z-axis direction, even at 10 • angle of attack despite the strong vertical flow component ( figure 10). This consistency occurs because airflow from the upper and lower wing surfaces are free to mix and equalize pressures in the gap. Thus we highlight the general pattern by plotting pressure coefficients along the chord line of the gap face-that is, at the intersection of the mid-plane across the gap and the root face of the gap, shown as a dotted white line in figure 10 (E). Figure 11 shows representative velocity fields and pressure coefficients measured over the SI-aileron wing deflected 4.7 • and 15.3 • upwards, at 2 • angle of attack. While we simulate the SI-aileron wing at a larger range of deflections, these two cases demonstrate the representative behavior at small and large aileron deflections. In both deflection cases, the pressure coefficients are generally negative on the lower surface and positive on the upper surface of the aileron, indicating a positive hinge moment that pushes the aileron down towards the neutral position. There are noticeable pressure spikes near the hinge axis at 75% chord, which is typical for ailerons due to the abrupt change in flow direction [34]. Greater aileron deflections lead to larger pressure spikes and higher hinge moments, which in turn required higher actuation work (equation (3)). Similar pressure distributions over ailerons have been observed in other studies [34][35][36].

Work requirements of the gapped wing
With an initial understanding of the steady flows over the gapped wing and aileron, we plot the work requirements (calculated by equations (1) and (3)) against rolling moment coefficient ( figure 12). This figure represents the work required by each control surface to produce a given rolling moment coefficient, via its main method of producing roll. The SI-gapped wing produces roll via change in angle of attack [12], while the SI-aileron wing produces roll with deflection angle [14]. Given that the two control surfaces produce roll by very different means, rolling moment coefficient provides a readily available common parameter, without requiring the development of a new performance metric. It allows us to quantify and compare how much roll each device was able to produce, without comparing the specific method of actuation. Rolling moment coefficient is simultaneously an appropriate indicator of control performance for any roll control surface, and an important parameter in roll control surface design [20]. Due to the novelty of the gapped wings, there is no current standard for comparing a deflecting and a planar control surface. However, this method of comparing roll and actuation work was used in previous work [15]. To vary rolling moment coefficient, we simulated the gapped wing across angle of attacks from 0 • to 12 • and the aileron deflected from −0 • to −30 • at a constant 2 • angle of attack.
Counter to what we expect, the gapped wing actuation work is higher than that of the aileron for low rolling moment coefficients ( figure 12). Both wings have a nonlinear relationship between work and rolling moment coefficient. However, at zero rolling moment coefficient, the aileron requires zero work while the gapped wing requires a nonzero baseline work. This is because it always takes work to actuate the gaps regardless of whether they produce roll. The NACA 0012 is a symmetric airfoil, meaning that at 0 • angle of attack, zero lift is produced by either semispan and thus zero rolling moment is produced. At any other angle of attack, exerting work to actuate the gapped wing results in a rolling moment. In fact, the gapped wing work curve is shallower than the aileron work curve. The curves intersect at a rolling moment coefficient of 0.0182, indicating that the gapped wing requires less work than the aileron wing for rolling moment coefficients above this point.
Note that the CFD simulations of the SI-gapped wing may predict higher actuation work than expected for both methodological and fluid dynamics reasons. Firstly, it is challenging to resolve flows in RANS simulations in the vicinity of small geometric features such as the gaps. As a result, there may be large errors associated with the gap force predicted by CFD simulations. RANS provides an appropriate compromise between accuracy and cost for this Figure 12. The SI-gapped wing requires less work than the SI-aileron wing to achieve rolling moment coefficients above 0.0182 and produces a higher maximum coefficient. The aileron was simulated at 2 • angle of attack and deflection angles from 0 • to −30 • , and the gapped wing was simulated across angles of attack from 0 • to 12 • . The gray data points are stalled and the gray '×' indicates the partially converged SI-gapped wing data point. The SI-gapped wing values depend on the area ratio R. study, and yields good agreement between simulated and experimental results. However, future work could employ higher fidelity simulation tools to confirm the work estimates and resolve finer details in the flow. The choice of turbulence model could further affect the work predictions [23]. The higher work of the gapped wing at lower rolling moment coefficients could also be due to the fluid mechanics in the gaps. The flow through the gaps creates a stronger force than the airflow over the aileron at lower rolling moment coefficients. However, at higher rolling moment coefficients, the greater magnitudes of the pressure coefficients over the aileron lead to a larger hinge moment on the aileron than the suction force on the gaps. As the rolling moment coefficient increases, the work required to actuate the gaps does not increase as quickly as the work required to actuate the aileron, leading to the flatter gapped wing work curve.
The gapped wing and aileron wing also have different roll control behaviors post-stall ( figure 12). The aileron work curve exhibits a nearly vertical jump at a rolling moment coefficient of 0.0237, indicating that the aileron is likely partially stalled above this rolling moment coefficient: any increase in deflection angle leads to a slight increase in rolling moment coefficient but a large penalty in work requirements. This result may be because the aileron work curve is taken at 2 • angle of attack. Thus, the semi-span with the aileron stalls because of the high deflection angles, while the clean semi-span does not stall due to the low angle of attack. In this situation the wing continues to produce a positive rolling moment because the clean semi-span still produces more lift than the semi-span with the stalled aileron. Conversely, the gapped wing reaches a higher maximum rolling moment coefficient of 0.0246 before stalling. At this point, the rolling moment coefficient drops drastically while actuation work remains constant. Here, both semi-spans of the gapped wing are at a high angle of attack of 12 • , so the flow over the whole wing is likely separated. The gaps are completely enveloped by this separated flow, meaning their presence has little bearing on the forces and moments produced by the wing. That is, the stalled gapped wing behaves like a wing without gaps and produces very small rolling moments. Recall that the SI-gapped wing data point at 12 • angle of attack is partially converged (the gray '×' in figure 12). This point is included to demonstrate the qualitative post-stall behavior of the SIgapped wing. Figure 12 suggests that the gapped wing reaches a greater maximum rolling moment coefficient than the aileron before stalling. From the experimental validation (figure 5), we know that the simulations under-predict the maximum rolling moment coefficient of the SI-gapped wing [12]. It is likely that the simulations under-predict the maximum rolling moment coefficient of the SI-aileron wing as well. Despite these conservative predictions, figure 12 shows that the gapped wing still achieves a higher rolling moment coefficient than the aileron wing before stall. The SI-gapped wing produces a maximum rolling moment coefficient of 0.0246 at 10 • angle of attack ( figure 12). This value is equivalent to the aileron deflected more than 18.3 • , at which point the aileron has already partially stalled. Further, recall that the aileron is plotted at a low angle of attack of 2 • in figure 12. It is possible that the aileron may stall at lower deflections when at a higher angle of attack like the gapped wing [14,20]. In these scenarios, the aileron wing would produce an even lower maximum rolling moment coefficient, because aileron performance is generally proportional to deflection angle but constant across angle of attack [14]. Therefore, the SIgapped wing could likely produce greater maximum rolling moment coefficients for less work compared to the SI-aileron wing. In an experimental setting, we expect the aileron and gapped wing work curves would continue on similar pre-stall trends to even higher rolling moment coefficients than found using CFD. Given this advantage, it is important to note that the gapped wing loses control effectiveness after stall, while the aileron wing retains some effectiveness at the simulated 2 • angle of attack due to its partial stall.
The dependence of the gapped wing's control effectiveness on angle of attack could make implementing the gaps more complex. Since the gapped wings are less effective at lower lift coefficients, they may need to be augmented with an additional control surface at those flight conditions. For example, at low angles of attack, roll could be primarily controlled by an aileron with the gaps assuming control at higher angles to lower work costs and delay stall. Alternatively, a hybrid control surface could take the form of an aileron with gaps along its surface. The gaps could open at higher aileron deflections to delay stall and remain closed at lower deflections to take advantage of lower aileron work costs. Further investigation is needed to determine the validity of this concept. The gapped wings also could require a complex controller that is able to manage the changing control surface effectiveness across angle of attack. Similarly, the gapped wing could be paired with an instantaneous pitch-up maneuver before actuation, to capture maximum rolling performance at the high lift coefficient [37]. This approach would assume that the gapped wing is necessarily changing angle of attack before rolling, so including the work to perform the pitchup maneuver could provide a more complete picture of the design space.
These considerations show that there is room to optimize the gaps to improve their practicality. Future work will analyze the effects of gap geometry on aerodynamic behavior, including gap size, number, and position. For example, we hypothesize that shortening the gaps and starting them further aft along the chord could reduce the actuation work. Shortening the gaps could effectively decrease the area exposed to the suction pressures in the gap. The more aft-ward position could also reduce the overall suction force on the gaps, since the pressure coefficient increases towards the trailing edge. However, this change may also lower the maximum rolling moment coefficient achievable by the gaps. Thus, we speculate that the length of the gap would need to balance actuation costs with the desired maximum rolling moment coefficient. Higher-fidelity CFD models such as large eddy simulation (LES) could also be used to resolve smaller and more complete flow structures and better understand the stall behavior of the gapped wing. Performing unsteady simulations with LES, detached eddy simulation, or unsteady RANS would also provide a more comprehensive understanding of the fluid mechanics of the gapped wings, particularly at high lift coefficients. Additionally, future work will focus on developing new methods for comparing the gapped wings with existing control surfaces. Due to the novelty of the gapped wing, we did not focus here on devising new performance metrics. However, it would be valuable to compare the gapped wing and aileron using different parameters or control effectiveness metrics that have yet to be invented. Finally, future work will compare the gapped wing to a more realistic wing with two ailerons that deflect differentially, to provide a holistic perspective of gapped wing roll control performance.

Conclusion
Here, we performed the first CFD simulations of flow over a novel whiffling-inspired gapped wing. We experimentally validated the simulations, visualized important aerodynamic mechanisms, estimated the actuation work requirements of a gapped wing, and compared the results to those of a conventional aileron. The CFD data agree well with previous experimental data of both a gapped wing and conventional aileron, building confidence in our results [12,14]. The novel flow visualization results also provide insight into control effectiveness, work requirements, and overall aerodynamic behavior of the gapped wings. For example, the gaps re-energize the boundary layer over the suction side of the trailing edge, which may help delay stall. Additionally, each gap produces a pair of vortices. Except for a large central gap vortex, the gap vortices are more distributed along the semi-span than the aileron vortices. This unique vortex arrangement contributes to a favorable lift distribution, such that the gapped wing produces comparable roll and less yaw than the aileron. The vortices may also be a lead cause of the gapped wing's varying roll effectiveness across angle of attack. The simulations further show that flow within a gap itself is characterized by a recirculation region that causes negative pressure coefficients on the majority of the gap face. The negative pressure distribution indicates a suction force, requiring work to hold the gaps open. The magnitudes of the pressure coefficients increase with angle of attack, leading to a corresponding increase in gap cover work requirements. Furthermore, the actuation work requirements for both the gapped wing and the aileron wing are based directly on aerodynamic loading, a new approach than previously taken. A comparison of the control surfaces' work and rolling moment coefficients demonstrated that the gapped wing requires less work for rolling moment coefficients above 0.0182 and produces a greater maximum rolling moment coefficient. However, the gapped wing loses control effectiveness poststall. Since the gapped wing effectiveness varies across angle of attack, it may require a more complex controller or augmentation by another control surface at low rolling moments. Overall, the CFD simulations indicate that the gapped wings could be an alternative method of roll control for energy-constrained aircraft when higher rolling moment coefficients are required.
Both the previous experimental study [12] and the current work investigated a baseline SI wing without gaps (SI-baseline wing) as a comparison to the gapped wing. Here, we experimentally validate the SIbaseline wing. We also compare the pressure coefficients at a gap section with pressure coefficients at a baseline wing section. Note that differences in aerodynamic forces and moments between the SI-baseline wing and SI-gapped wing are thoroughly discussed in the previous experimental work, and are not repeated here [12].
The CFD results of the SI-baseline wing agree well with the previous experimental data [12] (appendix figure 13). This matching is expected, given the good agreement between CFD and experimental data for both the SI-gapped wing (figure 5) and JH-aileron wing ( figure 6). The baseline wing was simulated as an SI-aileron wing with a 0 • aileron deflection, at angles of attack ranging from 0 • to 10 • .
Appendix figure 14 compares the pressure coefficients along the gap with the pressure coefficients over a baseline wing section. The baseline pressure distribution appears nominal for a NACA 0012 section. Comparatively, the pressure coefficients along the chord in the central gap fluctuate to local maxima and minima before returning to a near-nominal value at the trailing edge. The baseline wing reaches a slightly higher pressure coefficient at the trailing edge than the gapped wing. The pressure coefficients along the gap (yellow line) are the same as those plotted in figure 10(B), measured along the chord line of the Figure A13. The SI-baseline wing CFD data agreed well with the previous experimental results. The baseline wing is simulated as the SI-aileron wing with a 0 • deflection at angles of attack from 0 • to 10 • . Figure A14. Compared to the pressure coefficients over the SI-baseline wing (brown), the pressure coefficients over the SI-gapped wing (yellow) oscillate to local minima and maxima. The SI-baseline wing also experiences a slightly higher pressure coefficient at the trailing edge than the SI-gapped wing. root face of the central gap. The pressure coefficients of the baseline section (dark brown line) were measured from the SI-gapped wing at the wing root, away from the gaps.

Appendix B. Estimating the boundary layer over the wing
This appendix describes the process by which we estimate relevant boundary layer properties of the SIgapped wing. We evaluated the boundary layer at 95% of the wing chord at two spanwise locations, as shown with the black capped lines in figure 7. This chord location enabled us to investigate the boundary layer near the trailing edge, where it was thicker and the effects of the gaps were more readily apparent. The two spanwise locations included the wing root (over the baseline wing section) and 0.0254 mm inboard of the root face of the central gap (in the vicinity of a gap). To estimate the thickness of the boundary layer, we first needed to determine the typical velocities at the desired locations, but in the absence of viscosity [19]. Finding these inviscid velocity magnitudes was necessary because the wing surface was curved, as opposed to a simple flat plate [19]. We ran an inviscid simulation of the SI-gapped wing at 10 • angle of attack and the same flow properties as shown for the SI-gapped wing in table 1. As before, we performed the RANS simulation in STAR-CCM+, using the inviscid and coupled solvers. We removed the prism layers over the wing surface since there was no boundary layer for them to capture, resulting in a mesh with 1.52 million cells. The simulation was terminated when the residuals of x-momentum, y-momentum, z-momentum, and continuity fell below 1 × 10 −4 , which occurred after 124 iterations. We then measured the velocity magnitude on the upper and lower surfaces at the two desired locations. Smoothed values (interpolated to the nodes) calculated internally by STAR-CCM+ were used. At about 95% chord, the velocity magnitude over the upper surface was 15.3 m s −1 in the vicinity of the gap, and 14.8 m s −1 over the baseline section. These values were close, but not equal, to the freestream velocity due to the shape of the airfoil [19].
We then found the height of the boundary layer at the two locations in the inviscid simulation. The height of the boundary layer is often defined as the point at which the velocity magnitude reaches 99% of the inviscid velocity magnitude [19]. This height is known as the 99% boundary layer thickness [19]. We found the 99% boundary layer thickness to be 0.010 m at 95% chord over the baseline section, and 0.026 m at 95% chord in the vicinity of the gap, measured normal to the wing's surface at each point. Concurrently, we measured the velocity magnitude profiles normal to the upper surface of the wing at each point. Again, we used the smoothed node values of velocity magnitude that were internally calculated by STAR-CCM+. These boundary layer velocity profiles corroborated that the flow through the gap re-energized the local boundary layer (appendix figure 15). At 95% chord of the baseline section (dashed blue line in appendix figure 15), the velocity magnitude steadily and slowly increased to higher values as the wall distance increased. Conversely, the velocity magnitude at 95% chord near the gap (solid red line in appendix figure 15) quickly jumped to a higher value at a lower wall distance, then gradually increased. Below 11.6 m s −1 , the boundary layer near Figure A15. Boundary layer velocity profiles over the baseline section and in the vicinity of a gap highlight that the gap re-energized flow over the suction side of the trailing edge. Velocity magnitude was measured normal to the wing surface at 95% chord, at two span locations: the wing root and 0.0254 mm inboard of the root face of the central gap (shown in figure 7). the gap generally reached a given velocity at a lower wall distance than the boundary layer at the baseline section. As discussed with figure 7, this result indicated that a thinner portion of the near-gap boundary layer experienced slower velocities than the baseline boundary layer. Thus, the gap re-energized flow over the suction side of the trailing edge.