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Diese Studie untersucht die aerodynamische Leistung urbaner Luftfahrzeuge (UAM) und konzentriert sich dabei auf die Vorteile aerodynamischer Formoptimierung für Flügel mit dezentralem Antrieb. Die Forschung untersucht, wie sich die unterschiedliche Anzahl der Propeller auf die aerodynamische Effizienz und die Optimierungsergebnisse auswirkt. Schlüsselthemen sind die Auswirkungen der Interaktion zwischen Propellerflügeln auf Auftrieb und Zugkraft, der Optimierungsprozess mittels RANS-basierter Methoden und die Zielkonflikte zwischen verschiedenen Propellerkonfigurationen. Die Studie kommt zu dem Schluss, dass ein einflügeliger, an der Spitze montierter Propeller zwar die aerodynamische Effizienz signifikant verbessert, das Hinzufügen weiterer Propeller jedoch zu sinkenden Erträgen führt. Die optimierten Flügelkonstruktionen zeigen eine konsequente Reduzierung des Luftwiderstands um 6,5% in allen Konfigurationen, was das Potenzial für signifikante Verbesserungen der UAM-Fahrzeugleistung durch aerodynamische Formoptimierung unterstreicht.
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Abstract
Recent interest in urban and regional air mobility and the need to improve the aviation industry’s emissions has motivated research and development of novel propeller-driven vehicles. These vehicles range in configuration from conventional takeoff and landing designs to complex rotorcraft that transition between vertical and horizontal flight. These designs must be optimized to ensure optimal efficiency throughout their missions, leveraging the tightly coupled nature of propeller-wing interaction. In this work, we study the NASA tiltwing concept vehicle wing with varying numbers of propellers, ranging from no propellers to five propellers evenly spaced along the wing. Using aerodynamic shape optimization, we optimize the wing shapes for each propeller-wing configuration, minimizing the wing drag. These optimizations are carried out with DAFoam, a discrete adjoint implementation of OpenFOAM, embedded within OpenMDAO and the MPhys optimization framework. The optimizations show that the lowest drag configuration is a single propeller mounted at the wing tip. Increasing the number of propellers slightly increases drag compared to the single propeller configuration. However, aerodynamic shape optimization considering propeller-wing interaction yields a negligible benefit compared to aerodynamic optimization of an isolated wing that is subsequently trimmed to a desired flight condition in the presence of a propeller.
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1 Introduction
There is significant research and industry interest in urban air mobility (UAM), evident from the large number of electric vertical takeoff and landing (eVTOL) aircraft under development [1]. Many proposed designs feature propellers mounted on wings or vehicle appendages, using those propellers for both vertical and horizontal flight [2]. These complex configurations have motivated work on several important areas of technological development, ranging from noise and community acceptance to vehicle efficiency. The aerodynamic performance of these vehicles is critical to vehicle efficiency and low-emissions operation. This need for highly efficient vehicles has motivated novel research work on the topic. Several research efforts found that considering propeller-wing interaction provides more precise predictions that can be beneficial for improving aircraft performance [3‐9]. Even a small improvement in aerodynamic efficiency can significantly decrease the energy and power required for a UAM vehicle, and thus, optimizing all aerodynamic components of an aircraft has the potential to substantially improve a vehicle’s lifetime environmental cost. Accurately modeling aerodynamics is central to understanding a baseline design’s performance and optimizing to improve aerodynamic efficiency.
There are numerous ways to model the aerodynamics of propeller-wing interaction, with varying levels of fidelity. For conceptual and preliminary design analyses, the vortex-lattice method coupled with blade element momentum theory provides a low-cost and low-fidelity result [8, 10‐14]. Higher-fidelity methods leveraging computational fluid dynamics (CFD) such as the Euler or Reynolds-averaged Navier–Stokes (RANS) equations can yield more accurate results, with a variety of computational costs [8, 15‐21].
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An effective method to capture time-averaged influence and performance is to simulate the propeller as an actuator disc, adding axial and tangential velocity into the domain. Adding time-averaged influence to the flow field can be done in multiple ways. One method is with a fixed actuator disc profile. This approach adds axial and tangential forces into the flow field to mimic the propeller; the axial and tangential force distributions are constant and based only on the required thrust, diameter, and actuator model specific tuning parameters. Alternatively, the actuator disc model can be coupled with low-fidelity blade element momentum theory (BEMT) and the Euler or RANS equations to improve the prediction of the propeller and wing performance. In this method, the forces imposed on the flow field are computed by BEMT using the local flow field computed in the higher fidelity simulation. The computed forces are then applied to the flowfield. [8, 15‐20]. To accurately capture the unsteady effects of propeller-wing interaction, the propeller can be simulated as an actuator line using unsteady RANS [15, 17]. To capture the propeller and wing performance accurately, they can be simulated together with fully resolved geometries [15, 17, 20, 21]. This approach requires unsteady RANS or large-eddy simulation. While this provides the highest-fidelity result, it incurs significant computational cost. Instead, a mid-fidelity actuator disc model can accurately account for time-averaged propeller effects with a manageable computational cost.
Optimization considering propeller-wing interaction has been studied with a variety of fidelities and approaches. Fundamental work with reduced fidelity models found that an optimal lift distribution for a wing influenced by a propeller may differ from the conventional optimal design matching an elliptical distribution [4]. This study also compared inboard-up, outboard-up, and counter-rotating propellers in both tractor and pusher configurations. Velduis and Heyma [6] expanded on this finding, optimizing a wing for minimum drag with a lift constraint, modifying only the twist of the wing. The study concluded that the wing should have a higher twist behind the down-going side of the propeller disc and a lower twist behind the up-going side. The wing was also optimized without a propeller and then simulated with both inboard-up and outboard-up configurations, finding that inboard-up improves performance more than the outboard-up case. A similar study was carried out with chord and twist variables as well, using similar lower-fidelity methods [22].
In addition to studies focused solely on propeller-wing interaction, multidisciplinary design optimization (MDO) has been applied to vehicle design problems considering this phenomenon. Alba et al. [10] performed MDO of a wing’s planform, considering weight and performance, as well as propeller-wing interaction. Other MDO efforts leveraged lower-fidelity methods to optimize mission performance, including aerodynamic, structural, and acoustic considerations [11, 23]. Recent work has focused on applying higher-fidelity CFD to analyze propeller-wing interaction, for example, studying a propeller in a tractor configuration using an actuator disc coupled with loads computed using blade element theory [24]. Similar work using gradient-based optimization implemented an actuator disc model with fixed axial and tangential influence on the flow field to optimize a wing considering a variety of propeller location and wing design variables [25, 26].
Propeller-wing interaction is central to understanding the flow physics of propeller-driven UAM vehicles. While a variety of low- to high-fidelity techniques exist for modeling this phenomenon, actuator disc-based analysis has been demonstrated to be an accurate and cost-effective methodology for understanding time-averaged physics. Such models have been used for RANS-based optimization considering one- and two-propeller configurations; however, they have not been applied to existing vehicle designs.
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NASA has provided a fleet of conceptual air vehicle designs as test platforms for research and further development [27‐29]. These concept vehicles, ranging from quadrotor to tiltwing configurations, resemble the UAM vehicles under development in industry.
In this paper, we use the NASA tiltwing concept vehicle that features six propellers across the main wing with two propellers on the horizontal tail [29]. We use an actuator-disc-based model in the wing optimization, considering a varying number of propellers, from no propeller to five propellers. We investigate the effect of the number of propulsors on a wing in cruise flight and the advantages and disadvantages of each configuration. Additionally, this work quantifies the importance of modeling propeller-wing interaction when performing aerodynamic shape optimization of distributed propulsion configurations.
2 Computational methods
Aerodynamic optimization requires a single discipline solver to compute the aerodynamic performance of a given aircraft geometry. This solver is run at each design iteration with an updated geometry to understand the performance at each step. The flow of data through one iteration of a RANS-based aerodynamic optimization is shown in Fig. 1 using an extended design structure matrix (XDSM) [30]. The green blocks represent analysis components, the red block represents an implicit component that must be converged, the blue block represents the optimizer, and the gray blocks denote data passing from one component to another. The optimization starts with a pre-processing stage in which a baseline design is provided and parametrized. This baseline design consists of the geometry and the operating conditions passed to the optimizer. In this pre-processing step, the geometry is parametrized so that it can be updated to deform its shape for each optimization iteration.
Once the pre-processing is complete, the optimizer repeatedly carries out aerodynamic analysis and gradient computation. The geometric shape is updated at each optimization iteration, yielding aerodynamic surface mesh deformations that propagate through the volume mesh. This volume mesh is provided to the aerodynamic solver, which computes an updated flow field. The flow field is then used to perform function evaluations to find quantities of interest, such as lift and drag. Once these quantities are calculated, gradients are computed using the adjoint method to provide the derivatives needed for gradient-based optimization efficiently. This optimization loop is carried out repeatedly until the optimizer converges to an optimal design.
The tools used in this work were developed as part of the MDO for Aircraft Configurations at High-fidelity (MACH) framework [31, 32] and are well suited for aerodynamic shape optimization [33‐35]. The MACH framework is integrated within OpenMDAO [36], a framework designed for multidisciplinary gradient-based optimization, and efficiently handles the couplings between computational solvers using a toolkit called MPhys.1 MPhys consists of a set of standards and interface components that streamline the process of coupling high-fidelity optimization tools. The geometry engine, mesh deformation tool, aerodynamic solver, and optimizer, along with the gradient computation methodology, are described below.
2.1 Geometry manipulation
At each iteration, the optimizer provides updated design variables that deform the geometry surface, moving towards an improved design. Within the MPhys framework, pyGeo is a flexible geometry engine that is well suited for problems featuring lifting surfaces such as wings [37]. While pyGeo can also be used to generate wing geometries using input parameters such as airfoil shapes, wingspan, twist, and dihedral, it is particularly useful for parametrizing and updating existing geometries. The pyGeo tool utilizes free-form deformation (FFD), a geometry manipulation technique originally developed for computer graphics [38]. A geometry within an FFD grid is considered a flexible material that can be locally or globally compressed, stretched, rotated, or translated to morph its shape. For aerodynamic analysis and optimization, pyGeo and FFD methods deform the geometry at each iteration based on the updated design variables provided by the optimizer. This deformed shape is then passed to the mesh deformation tool to warp the aerodynamic volume mesh.
2.2 Mesh deformation
Once the geometry being optimized is updated during an iteration, the associated aerodynamic volume mesh must be updated. For aerodynamic optimization, this update can be performed by warping the volume mesh surrounding the updated geometry. IDWarp is a tool integrated into the MPhys toolkit that implements an inverse distance weighting algorithm to deform a volume mesh [39]. IDWarp is mesh-type agnostic, meaning it is flexible for both structured and unstructured mesh types. It is also efficient, meaning it requires a small fraction of the time needed for aerodynamic analysis. Once the volume mesh is warped to match the updated geometry, the mesh is passed to the aerodynamic solver to converge the flow field for that iteration.
2.3 Aerodynamic solver
DAFoam is a discrete adjoint implementation for OpenFOAM ESI v1812 [40] integrated into MPhys as a RANS aerodynamic solver. A modified version of the RhoSimpleFoam steady-state solver is used for this work, with wall functions and a Spalart–Allmaras turbulence model [41‐43]. While DAFoam does include other turbulence models, such as the k-\(\omega\) SST model, the Spalart–Allmaras model is used for its simplicity and improved gradient accuracy. This tool leverages the adjoint method and algorithmic differentiation to enable unstructured-mesh aerodynamic analysis and optimization. DAFoam can be used for single or multipoint optimization, considering a variety of functions of interest as objectives and constraints. DAFoam has been used extensively for both single- and coupled-discipline optimization cases such as propeller-wing interaction optimization, vehicle aerodynamic optimization, and turbine blade optimization [26, 40, 44]. DAFoam has also been modified to allow for propeller modeling using an actuator disc formulation that applies equal and opposite time-averaged propeller forces to specified volume cells [18]. This formulation uses an analytic model [45] to obtain the radial distribution of the axial and tangential forces on the propeller region. The model is modified for optimization to smooth the numerical instabilities generated by discontinuities in the propeller force distributions [26]. The actuator disc model is based on a standardized, non-dimensional radius,
where r is the radial distance from the axis of rotation, \(r_{in}\) is the root cut radius of the propeller, and R is the outer radius of the propeller. The radial distribution of the axial force per unit radius is written as
The value of \(\tilde{F}\) is adjusted such that the axial force matches the total thrust requirement. The parameters controlling the shape of the thrust profile are set to \(m=1\), \(n=0.5\), and \(a=1\). Increasing m and decreasing n shifts the position of maximum thrust towards the tip. Using a value of a greater than 1.0 results in a finite load at the tip or outer radius. The radial distribution of the tangential force per unit radius was modeled using the following equation:
where P/D is the pitch-to-diameter ratio, \(\beta = \varepsilon /\left( R-r_{in}\right)\) is the smoothness control parameter, and \(\epsilon\) is the mesh cell size in the actuator region. The second term in the denominator adds a finite radius even at the location where r = 0 to prevent the tangential force from going to infinity.
The axial and tangential force distributions on the nominal tiltwing vehicle propeller were not quantified and thus not available to inform the actuator disc model parameters used in this study. Instead, the parameters used are the same as in previous design optimizations with the same actuator disc model, with the exception of n, which was adjusted to move the peak propeller loading inboard [45]. These values were obtained by comparing them against the computationally simulated and experimentally tested PROWIM propeller [25, 26]. The force distributions are scaled to provide the prescribed total thrust for the tiltwing.
DAFoam converges the flow field, accounting for the actuator disc effects, and computes quantities of interest, such as lift and drag on the geometry. Additionally, DAFoam computes the gradients needed for gradient-based optimization using the matrix-free adjoint method [46].
2.4 Optimizer and gradient computation
This work utilizes SNOPT integrated within pyOptSparse, a package for gradient-based optimization of large, sparse problems [47]. SNOPT handles both feasibility and optimality, as well as linear and nonlinear constraints independently, guaranteeing the feasibility of the linear constraints once a feasible region in the design space has been found [48]. SNOPT and pyOptSparse are well suited for aerodynamic optimization, which features many design variables and a few objectives and constraints. DAFoam is the only solver used in this study. First, the RhoSimpleFoam solver wrapped within DAFoam converges the flow field. Second, the adjoint solver within DAFoam computes model derivatives using the adjoint method based on the converged flow field. This adjoint computation involves the geometric design variables, \(\textbf{x}\), the aerodynamic state variables, \(\textbf{u}\), the function values of interest, f, and the residual values \(\textbf{r}\). The function of interest and residual values are dependent on both the design and state variables as
where the residual is assumed to be converged. Applying the chain rule to differentiate these equations with respect to the design variable vector and following the derivation of the adjoint method to combine the two equations [49, Sec. 6.7], the total derivative of a function of interest can be written as
which can then be substituted into the total derivative equation. Using this equation, we no longer need to invert a matrix. Instead, we solve the system once for a given f. This means that the cost of computing derivatives is dependent only on the number of outputs, not the number of inputs.
The adjoint method is synonymous with reverse-mode derivative computation because the two methods are typically implemented together. Both techniques are most efficient for gradient computation problems in which there are more inputs than outputs because the cost of the adjoint and reverse-mode derivatives scale with the number of output variables. This characteristic is particularly useful for aerodynamic shape optimization in which there are typically few outputs, such as lift and drag, but many inputs, such as shape design variables. The DAFoam adjoint implementation and the explicit derivatives from the associated geometry and mesh warping components are computed for each optimization iteration, providing gradients to the optimizer to determine the design variable values for the subsequent iteration.
3 Case setup
The NASA tiltwing vehicle studied in this work features several flight states spanning hover flight with its propellers angled vertically to cruise flight with its propellers angled horizontally. Figure 2 shows renderings of the vehicle, where Fig. 2a shows the hover configuration and Fig. 2b shows the cruise configuration. We only use the cruise configuration in this work. The vehicle’s conceptual design includes six propellers spanning the main wing and two propellers mounted on the horizontal tail. In its cruise configuration, the vehicle operates with the wing and propellers tilted forward for horizontal flight. All propellers operate at the same thrust. During cruise, the vehicle operates at 3,048 m at a speed of 79.74 m/s. The additional parameters governing this flight condition are shown in Table 1.
Fig. 2
NASA tiltwing concept vehicle. a Hover flight configuration. b Cruise flight configuration. [29]
Tiltwing concept vehicle cruise flight parameters at 3048 m
Flow parameter
Value
\(\rho _\infty\)
0.9049 kg/m\(^3\)
\(U_\infty\)
79.74 m/s
\(p_\infty\)
69692.1456 Pa
\(T_\infty\)
268.35 K
To study the effect of varying the number of propellers on the vehicle, the wing is analyzed and then optimized for the cruise flight condition. Figure 3 shows various configurations, ranging from no propeller to five propellers. For each configuration, the propellers provide an equivalent total thrust of 1041.27 N. All propellers operate at a constant pitch-to-diameter ratio of \(P/D = 2.74\). This value closely matches the advance ratio prescribed for the baseline vehicle design [29].
Fig. 3
NASA tiltwing concept vehicle wing with a varying number of propellers. a No propellers. b 1 propeller. c 2 propellers. d 3 propellers. e 4 propellers. f 5 propellers
The location and size of each propeller are dictated by the tiltwing vehicle’s nominal design and installation requirements. The cases with one and two propellers show large disc areas relative to the wing. This configuration would incur a weight penalty for the structure required to support such large propellers and the corresponding nacelles encasing the supporting propulsion systems. This weight penalty is not quantified because it is out of the scope of this study. Instead, we focus solely on aerodynamic performance.
All of the propellers rotate in outboard-down orientation and are 0.83 m in front of the wing leading edge, matching the baseline design [29]. For each case with a propeller, one propeller is mounted at the wing tip where the wing tip propeller of the nominal design is located. The additional propellers are mounted slightly below the wing, with the same constant vertical location as the nominal design. The spanwise locations of these propellers are interpolated in the available space on the wing, ensuring that the inboard end of the inboard propeller disc is tangent to 18% span along the wing, away from the fuselage. This study focuses on aerodynamic performance alone, so the propeller is placed at the wing tip following previous work by Koyuncuoglu and He [26], which shows that the aerodynamically optimal location for a propeller with an outboard-down rotation direction is at the wing tip.
The configuration dictates the radius of each propeller, ensuring that the discs are tangent to each other as in the nominal design. The tiltwing was initially designed with propellers operating tangent to each other. This configuration can be suboptimal for aerodynamic and aeroacoustic performance due to the high-speed flow interactions at each blade tip where the propellers are tangent. This study does not investigate the spacing between the propellers and does not optimize either the aerodynamic or aeroacoustic performance of the propellers and their interaction. The root cut radius is maintained as 20% of the propeller radius for each configuration, but the propeller hubs and root cutouts are not modeled. The single propeller case has an additional installation constraint because a radius that is too large would cause the propeller to intersect the wing. For this reason, the single propeller case does not occupy all of the allowable space on the wing, while the other configurations do. The propeller radius, thrust, and torque settings based on a constant pitch-to-diameter ratio are shown in Table 2 and are held constant throughout the entire optimization.
Table 2
Propeller operating conditions
Number of propellers
Radius (m)
Thrust (N)
Torque (Nm)
1
2.23
1041.27
1377.54
2
1.84
520.63
688.88
3
1.10
347.09
461.26
4
0.79
260.32
349.19
5
0.61
208.25
282.62
To accurately model the effect of the propellers, the aerodynamic mesh surrounding the wing in the vicinity and wake of the propellers must be fine enough. The mesh overlapping with the actuator zone must be fine enough to capture the actuator forces adequately added to the flow. A single mesh was used for all cases to ensure the cases were run with a standard mesh that did not introduce deviation and inaccuracy between configurations. This mesh has a large refinement zone that captures the entire volume in which a propeller may exist. The mesh refinement zones are shown in Fig. 4. The mesh refinement consists of two zones: a fine zone that extends one-half the maximum blade radius in each axial direction, forward and aft of the propeller hubs, and a coarse zone that extends from the aft of the fine zone to three times the maximum blade radius behind the blade hubs in the axial direction.
Fig. 4
Wing and propeller geometries encapsulated by the fine (red) and coarse (blue) mesh refinement regions. (Color figure online)
The aerodynamic mesh is generated using Pointwise and its T-Rex hybrid unstructured meshing algorithm.2 This method generates a semi-structured surface topology on critical areas of the mesh, such as the wing’s leading and trailing edges. It builds an ordered boundary layer mesh that grows from the surface until reaching isotropy. The volume mesh is then refined in the actuator zone regions, satisfying the specified cell edge lengths. Three meshes with varying levels of refinement were generated to understand the effect of mesh refinement on this study. The parameters of the meshes are detailed in Table 3, including the number of volume cells, the mean \(y^+\) value, and the refinement zone edge lengths. The refinement zones focus on the propellers, wing, and downstream wake to maintain a reasonable cell count and computational cost. The surrounding mesh outside the refinement zones is interpolated to smooth the transition between the finer and coarser mesh regions. This interpolation adds refinement upstream of the propeller discs, where the propeller slipstreams form. The meshes are generated with the intention of approximately doubling the refinement between each level, decreasing the surface and refinement zone line spacing by half with each level of refinement. The \(y^+\) values for the level 1 and level 2 meshes are in the buffer layer, where the Spalart–Allmaras turbulence model can produce inaccurate results. However, the refinement study ensures that the model captures the general trends in quantities of interest within reasonable modeling accuracy.
Table 3
Mesh refinement study parameters
Level
Volume cells
Mean \(y^+\)
Fine (m)
Coarse (m)
1
52,162,308
8.3864
0.025
0.05
2
7,781,776
16.3774
0.05
0.1
3
1,491,852
33.3169
0.1
0.2
The results for each mesh level simulating the nominal three-propeller design are shown in Fig. 5, including the computed coefficients of lift and drag in Fig. 5a and b, respectively. The mesh refinement uses the \(-1/3\) power of volume cell count in the x-axis to observe the sensitivity in quantities of interest to the mesh resolution. While computational cost constraints prevented reaching fully mesh-converged values, both quantities exhibit approximately linear trends with mesh refinement. This linear behavior enables prediction of mesh-refined values and validates the mesh configurations used for subsequent analysis and optimization. The optimization employs the level 3 mesh to balance computational efficiency with sufficient accuracy.
Fig. 5
Mesh refinement study. a Lift coefficient (\(C_L\)), b Drag coefficient (\(C_D\))
The tiltwing vehicle’s baseline design features an untwisted wing with no dihedral angle and a 10\(^\circ\) sweep angle along the quarter-chord. The sectional airfoil shapes are based on the GA(W)-1 airfoil, modified to have 18% thickness-to-chord about the camber line. The wing has a semi-span of 6.66 m with root and tip chords of 1.02 m and 0.71 m, respectively. The wing is designed to provide lift to support trimmed flight at cruise with a lift coefficient of \(C_L = 0.67\) [29]. Because this simulation does not include the entire vehicle geometry, computing the full vehicle lift and drag from these results is not possible. The wing alone is analyzed by fixing the desired lift of the wing and thrust of the propellers. Fuselage effects are not considered in the simulations, and the wing is simulated with a symmetry boundary condition at the wing root.
Each configuration is analyzed and trimmed to provide the desired lift to understand the performance difference between the propeller configurations. This trim procedure is carried out by iteratively adjusting the angle of the wing while maintaining the location and angle of the propellers constant, keeping the discs perpendicular to the incoming flow. The wing incidence angle and resulting drag counts (\(C_D \cdot 10^{4}\)) are shown in Table 4 to compare between the various propeller configurations.
Table 4
Baseline wing design performance
Number of propellers
Drag counts
\(\% \Delta\) Drag counts
Incidence angle (°)
0
278.2
–
2.54
1
255.5
−8.16
2.33
2
263.7
−5.21
2.35
3
264.8
−4.82
2.30
4
265.8
−4.46
2.26
5
266.7
−4.13
2.22
The results show the benefits that the propellers provide. The drag coefficient for the wing alone is 22.27 drag counts higher than that for a single propeller. However, while a one-propeller configuration provides a significant improvement over the wing-alone case, the drag improvement decreases with an increasing number of propellers. This trend is further explained in Sect. 5.2. The required wing incidence angle decreases with increasing number of propellers. The wing-alone case requires the largest incidence angle to achieve the desired trim, and the angle decreases as the number of propellers increases. While propeller-wing interaction is an active area of research, this result is consistent with the expected benefits of distributed propulsion in a tractor configuration because the blown wing downstream of the propellers can more easily accomplish the desired trim.
The spanwise distributions of normalized lift, twist, and thickness are shown in Fig. 6, comparing the results of all the propeller configurations. The lift is normalized by dynamic pressure (\(1/2 \rho _\infty U_\infty ^2\)). Visualizing the normalized lift provides a clear comparison against a theoretically optimal elliptical lift distribution. The lift profiles are similar for all cases and follow the same trend, matching the expected distribution with a higher lift near the root of the wing, decreasing towards the tip. In each case, the propellers’ effect is noticeable along the span, with oscillations of increased and decreased lift compared to the baseline. The wing sections in the wakes behind the propellers are subject to increased axial flow. However, regions behind up-going blades have the added benefit of an increased angle of attack due to tangential flow, while the down-going blades have the opposite effect. Consequently, the normalized lift distributions show the number of propellers for each case with one oscillation corresponding to a single propeller. For the baseline design, both the twist and thickness-to-chord distributions are constant for the entire wing.
Fig. 6
Baseline distributions of normalized lift, deviation of normalized lift from the no-propeller case, twist, and thickness for all propeller-wing configurations
Contour plots of the pressure coefficient, computed as \((p - p_\infty ) / (1/2 \rho _\infty U_\infty ^2)\), over each of the propeller configurations are shown in Fig. 7. The contours show the baseline distributions for the no-propeller case and the difference in pressure coefficient for the propeller-wing configurations, compared to the no-propeller case. For all of the configurations, there is a significant drop in pressure at the trailing edge of the wing tip. This change in pressure is suspected to be a flow phenomenon partially induced by the vortex forming at the tip of the wing. Additionally, all of the cases show the expected pressure coefficient trends, with lower pressure on the leading edge decreasing towards the trailing edge. However, the pressure coefficient contours for each case do differ due to the influence of the propellers. The no-propeller case has a smooth pressure coefficient distribution over the top surface, while the propeller-wing cases show oscillations in distribution due to the propellers. This is particularly noticeable for the cases with many propellers, with oscillations in the pressure coefficient on the surface near the leading edge. As with the normalized lift distributions, this oscillation in pressure coefficient is likely due to the up-going and down-going propeller regions adding axial and tangential flow velocity along the wing.
Fig. 7
Contour plots of pressure coefficient on the suction side of the baseline wing for each propeller-wing configuration. a Shows the no-propeller case, while the cases with, b 1 propeller, c 2 propellers, d 3 propellers, e 4 propellers, and f 5 propellers show contours of the deviation of pressure coefficient relative to the no-propeller case
These analyses show the impact of propulsors on the baseline wing: there is a significant decrease in drag from a single propeller but a decreasing benefit with additional propellers. The lift distribution does affect the CFD, but does not affect the actuator zones. As explained in Sect. 5.2, this conclusion also holds for the optimized designs. The required incidence angle to achieve the desired trimmed lift coefficient decreases with increasing number of propellers. The no-propeller case requires the largest incidence angle, decreasing with each additional propeller, with the exception of the two propeller case. This is likely a consequence of added flow over the wing, which increases in uniformity and strength with an increased number of propellers. This blown wing effect helps to increase lift at decreased angles of attack. These results indicate that each propeller configuration has advantages and disadvantages. Aerodynamic shape optimization may be useful for refining the wing design, accounting for propeller influence, and improving wing efficiency.
5 Optimization
Each propeller-wing configuration is optimized for the cruise flight condition to understand its performance and quantify the benefit of aerodynamic shape optimization. These optimizations are performed with the framework described in Sect. 2.
5.1 Case setup
Aerodynamic shape optimization relies on geometric parametrization to iteratively morph a geometry to drive toward an improved design. This work utilizes an FFD-based approach to embed the baseline wing design in a grid of points that can each be adjusted to change the contained wing design. The FFD parametrization of the baseline wing design is shown in Fig. 8, including all of the spanwise and chordwise point locations. Each spanwise set of points can be moved together to twist the wing, while each chordwise point on a span section can be moved in the z direction to adjust the shape of the wing cross-section locally. Changing the number of propellers may affect the optimal wing planform. However, optimizing the planform is out of the scope of this work because this study does not investigate vehicle trim or trim stability, which would be impacted by planform changes. The wing chord, sweep, and span are fixed throughout the optimization. The motion of the FFD points is interpolated to the wing geometry and is updated at each optimization iteration.
Fig. 8
Geometric parametrization of the wing using an FFD grid consisting of eight spanwise sections and ten chordwise points. Each spanwise section can twist the wing while the chordwise points adjust the local airfoil shape
This work aims to optimize the wing shape for each of the varying propeller configurations. The objective function for these optimization cases is the drag coefficient while the wing is constrained to provide the required lift of \(C_L = 0.67\), as needed for trimmed cruise flight. Each configuration requires different lift coefficients due to weight changes. This work focuses on aerodynamic optimization, and quantifying weight changes is out of the scope of the work. Thus, the lift coefficient is the same for all configurations. The propellers are assumed to be mounted perpendicular to the freestream velocity regardless of the wing angle of incidence, and any force generated by the propellers in the lift direction is considered negligible compared to the wing lift.
The design variables for each configuration are the FFD twist and vertical displacements governing the wing shape. The number of FFD sections is decided by balancing the cost and robustness of the problem with the size of the design space. Too few sections lead to an artificially small design space, while too many sections lead to non-orthogonal design variables and large optimization that are difficult and expensive to converge. To balance these competing considerations, we use eight twist sections that rotate the airfoils at the respective spanwise locations and one hundred sixty control points to adjust the wing shape. The twist design variables are typically more impactful than the shape design variables when operating at low speeds. However, the shape design variables are included in the optimization to understand if there are nuances to an optimal wing design when considering propeller-wing interaction.
This is a single-point optimization carried out at the cruise flight condition without structural considerations, so additional geometric constraints are imposed on the problem to avoid unrealistic optimized geometries that would be unable to perform other flight segments. These constraints require that the total volume of the wing remain between its starting volume and three times its volume and the thickness throughout the wing remain between one-half and three times its initial thickness. These geometric constraints are intended to represent structural and packaging considerations that are not modeled in this optimization. Furthermore, the radius of the leading edge remains equal to or greater than the initial radius. An optimal wing design considering only the cruise condition would result in a sharp leading edge that would stall in other flight conditions such as climb or descent. Adding a leading edge radius constraint avoids a sharp leading edge and represents the requirement to fly at other flight conditions. Additionally, the leading edge and trailing edge FFD points are constrained to move in equal and opposite directions to limit shape-generated twist and avoid non-orthogonal design variables. The optimization problem is tabulated in Table 5, showing the objective function, design variables, and constraints.
Table 5
Aerodynamic cruise optimization formulation considering wing shape design variables. The objective is to minimize drag, \(C_D\), by varying twist and wing shape parameters constrained by lift and geometric considerations
Function/Variable
Unit
Description
Quantity
Minimize
\(C_D\)
–
Drag coefficient of the wing
\({\textbf {Total objectives}}\)
\({\textbf {1}}\)
by varying
\(-10 \le \gamma \le 10\)
deg
Twist of each FFD section
8
\(-1 \le \Delta {z} \le 1\)
m
Vertical displacement of FFD points
160
\({\textbf {Total design variables}}\)
\({\textbf {168}}\)
subject to
\({V_\text {bl}} \le V \le 3\cdot {V_\text {bl}}\)
m\(^3\)
Volume constraint
1
\(0.5\cdot {t_\text {bl}} \le t \le 3\cdot {t_\text {bl}}\)
One optimization was carried out for each propeller-wing configuration, using the same objective, design variables, and constraints. The objective function, design variables, and constraints were scaled to \(\mathcal {O}(1)\) to improve the problem scaling for the optimizer. Additionally, the initial design was trimmed to the required lift before beginning the optimization to ensure the optimizer started from a feasible design. The cases were each run using SNOPT with a feasibility and optimality tolerance of \(10^{-5}\) and an approximated function precision of \(10^{-8}\). Additionally, the optimizer was limited to 500 major iterations because it is expected that beyond this point, the design will not see significant improvement. Each case ran with 4 nodes on a high-performance cluster, using 32 processors per node with 5GB of RAM per processor. The cases used a total of 128 processors, 640 GB of RAM,3 for up to 120 hours each. The cases were run to completion, either terminating when reaching the maximum number of iterations or exiting as dictated by SNOPT.
5.2 Optimization results
The optimization convergence histories for the propeller configurations are shown in Fig. 9, including the optimality and feasibility computed by SNOPT, the objective function, \(C_D\), and lift constraint \(C_L\). The optimality and feasibility histories show the optimization converging for both values, decreasing the optimality for each case to below \(5.4 \cdot 10^{-4}\), or by over two orders of magnitude from the initial value of \(5.4 \cdot 10^{-2}\), and decreasing the feasibility significantly. Given a decrease in optimality greater than two orders of magnitude and significant improvement in feasibility for each configuration, the optimizations are deemed to have terminated successfully. The \(C_D\) results show similar optimization histories for each of the propeller configurations, albeit with different starting values. The majority of the drag reduction occurred within the first 100 iterations for all designs, with subsequent iterations yielding only marginal improvements. The \(C_L\) constraint shows a similar trend, with each propeller configuration beginning at the trimmed value before deviating slightly in the first iterations. Once the optimizer completes the first 100 iterations, the value of \(C_L\) appears constant for the remainder of the optimization, satisfying the lift constraint.
Fig. 9
Optimization convergence histories for all propeller-wing configurations, showing optimality, feasibility, drag coefficient, and lift coefficient
The improvement in \(C_D\) for each case is shown in Fig. 10, comparing the values and percentage decrease for each propeller-wing configuration. As with the baseline design, the no-propeller case shows the highest drag values, while the one propeller case features the lowest drag, with additional propellers decreasing the benefit of the propeller influence on the wing. The results show that each optimization decreased the drag by approximately 6.5%, with small incremental improvements with increasing number of propellers. This result shows the significant benefit that a wing tip propeller spinning with an outboard-down rotation direction can have on improving a wing design. The benefit of wing tip mounted propellers has been previously shown experimentally and computationally [7, 9, 26]. This is likely due to its significant effect on countering induced vorticity and decreasing induced drag. The effect is similar in principle to contra-rotating propellers—just as a rear contra-rotating propeller recovers energy from the front propeller’s swirl, placing a propeller in a wing’s vorticity can recover some of the energy that would otherwise be lost as induced drag. The propeller effectively acts as an energy recovery device by converting some rotational kinetic energy from the vortex into additional thrust. The propeller influence spreads the vorticity horizontally, acting like an extension to the wing span. This improvement diminishes substantially with added propellers, because propellers are added further inboard from the tip and each propeller’s effect on the flow field is diminished. When the propellers are added inboard of the wing tip, the effect of the propeller’s swirl is diminished because it does not reduce the induced drag related to the vorticity at the wing tip. While the radius of the single propeller is limited by the installation requirements on the wing, the smaller radius still provides substantial aerodynamic improvement. The effect of accommodating a larger propeller is unknown for this particular vehicle design, but when optimizing for wing drag a larger propeller is not necessarily optimal [26].
Fig. 10
Comparison of optimization results between propeller-wing configurations, showing baseline and optimized coefficients of drag (\(C_D\)) and percentage improvement
The spanwise distributions of normalized lift, twist, and thickness for the optimized designs are shown in Fig. 11. The lift is again normalized by the freestream dynamic pressure, \(1/2 \rho _\infty U_\infty ^2\), to compare against a theoretically ideal elliptical lift distribution. Similarly to the normalized lift of the baseline designs, the lift distributions feature oscillations corresponding to the propeller locations on the wing. Interestingly, the single propeller case features a lower normalized lift near the root of the wing compared to all of the other configurations, including the no-propeller case. This decreased lift is countered by an increase in lift near the tip, corresponding to the up-going region of the propeller. This result shows that the single propeller case can move the lift distribution outboard without incurring a significant penalty due to induced drag because the wing tip propeller counteracts the induced vorticity. This result corresponds with the expected theory, with a single propeller causing the optimal lift distribution to deviate from elliptical [4]. Unlike the single propeller case, the remaining propeller configurations blow over the entire wing. For these configurations, the optimization shifts the lift distributions inboard towards the root compared to the baseline cases.
Fig. 11
Optimized distributions of normalized lift, deviation of normalized lift from the no-propeller case, twist, and thickness for all propeller-wing configurations, showing nearly elliptical normalized lift distributions with a varying twist but similar thickness distributions
As in the baseline analyses, the distributions feature oscillations centered around each propeller location. These oscillations are also visible in the twist and thickness plots, where the twist for the no-propeller case is significantly smoother than the twist for the cases with more propellers. The single propeller case features a single large oscillation near the tip, while additional propellers show more oscillations with smaller amplitude. The optimized designs all exhibit increased twist near the wing tip, likely influenced by the tip vortex formation. Compared to the baseline design, these optimized configurations do not show significantly lower trailing edge pressure at the tip.
The thickness is greater than the baseline thickness, shown in Fig. 6, across the entire wing except the tip section. This result shows that the current optimization problem does not benefit from decreasing thickness and that the lower bound on the thickness design variable is not design-limiting. Furthermore, the thickness for each design is largely consistent, featuring four oscillations. One possible explanation for these results is that the optimal thickness distribution is consistent for all cases, regardless of the number of propellers. The other possible explanation is that the optimal design may benefit from additional FFD sections, allowing for more nuanced geometric changes along the wingspan. For example, added FFD sections behind each propeller could allow for increases and decreases in twist or thickness to more effectively capture local propeller upwash and downwash. The current number of FFD sections may not allow for such detailed refinement and may lead all of the cases to the same thickness distributions.
The study would benefit from increasing the number of spanwise FFD sections, but this can decrease the optimization robustness because closely spaced FFD sections increase non-orthogonality in the design variables. Though an updated parameterization may further improve results, these results show that the overall lift distribution for each configuration was optimized from the baseline to distribute lift more elliptically over the wing. Additionally, each configuration features a unique twist distribution with a largely consistent thickness distribution. This study did not investigate the importance of shape design freedom; optimizing the wing twist alone may be sufficient to obtain significant wing drag improvements. Further design freedom is needed to understand if the optimized wing could smooth the propeller-induced lift oscillations and further decrease overall drag.
The contours of the pressure coefficient over each of the propeller configurations are shown in Fig. 12. Compared to the baseline design results shown in Fig. 7, the optimized results do not feature a significant drop in pressure at the trailing edge of the wing tip. Instead, each configuration shows a smooth suction peak near the leading edge of the wing, decreasing towards the trailing edge without abrupt pressure changes. This result suggests that aerodynamic shape optimization can mitigate the flow phenomenon at the wing tip and likely better capture both the forming wing tip vortex and propeller wake without an inefficient pressure distribution over the surface. Similarly to the baseline results, the contours for each configuration show the effect of the propellers, with oscillations in the pressure coefficient on the leading edge of the wing. This effect corresponds with the spanwise lift distributions shown in Fig. 11, where the normalized lift distributions include oscillations even for the optimized designs.
Fig. 12
Contour plots of pressure coefficient on the suction side of the optimized wing for each propeller-wing configuration. a Shows the no-propeller case, while the cases with, b 1 propeller, c 2 propellers, d 3 propellers, e 4 propellers, and f 5 propellers show contours of the deviation of pressure coefficient relative to the no-propeller case
The optimized results show an approximately 6.5% improvement across all of the propeller configurations. This improvement is primarily driven by a redistribution of lift along the span of the wing, moving the lift inboard toward the root. The local oscillations due to propeller influence on each configuration were not smoothed and the optimized distributions show that the design space may need to be expanded further to improve the designs for cases with additional propellers. Ultimately, the results of both the baseline and optimized designs suggest that a single wing tip propeller provides the greatest improvement for optimal cruise performance.
5.3 Investigation of optimum
While it is possible to include propeller-wing interaction in aerodynamic shape optimization, it is not clear if this actually results in improved designs. In this study, the no propeller optimized design is simulated with propellers and trimmed to the desired flight condition. We use this to quantify the importance of considering propeller-wing interaction when performing RANS-based aerodynamic shape optimization. The propeller contribution was added to the flow field, and the wing was uniformly angled to trim to the required lift constraint. Figure 13 compares the drag between the optimized designs considering propeller-wing interaction and the no-propeller optimized design with propellers added.
Fig. 13
Difference in \(C_D\) between optimized designs considering propeller-wing interaction and no propeller optimized design with propellers added
Figure 13 shows the percentage difference in \(C_D\) between the optimized design considering propeller-wing interaction and the no propeller optimized design with propellers added. The resulting coefficients of drag are nearly identical for each case, with optimization considering propeller-wing interaction providing only a small fraction of a percent improvement over no propeller optimization. This result shows that there is nearly no benefit to capturing propeller-wing interaction in RANS-based aerodynamic shape optimization, regardless of the number of propellers, which is consistent with the findings of Chauhan and Martins [25].
The additional cost required to accurately model and capture propeller effects on the flow field near a wing is not justified for any of the configurations studied. This means that similar wing performance can be achieved without actuator zone models and with a less expensive mesh without the significant number of extra cells needed for the propeller refinement zone. This study confirms previous optimizations by Chauhan and Martins [25], showing that optimizing a wing alone and subsequently adding the desired number of propellers can yield a nearly optimal wing design. These results extend the previous optimizations to show that the conclusion is valid for configurations with multiple propellers.
6 Conclusion
This work studies the NASA tiltwing concept vehicle and optimizes the wing considering propeller-wing interaction in various configurations. This research combines RANS-based aerodynamic shape optimization with adjoint-based gradient computation to study how the number of propellers along a wing’s span can impact its drag in cruise flight. Additionally, this work describes the potential relative improvements for each configuration that are available with aerodynamic shape optimization.
The baseline analyses show that adding a single wing tip-mounted propeller has a notable effect on decreasing drag while adding additional propellers decreases this improvement. However, added propellers decrease the wing twist needed to achieve the desired lift, as expected with blown wings utilizing distributed propulsion. The optimized wing designs show a similar result, with the drag of each configuration decreasing 6.5%, maintaining the same trends as the baseline analyses. Additionally, increasing the number of propellers results in an optimized lift distribution that closely matches a conventional elliptical distribution, suggesting that a wing nearly entirely subject to propeller wakes matches conventional wing theory.
More design freedom, in the form of additional spanwise design points or planform variables, may further improve the optimized wing designs, allowing unique variations that adapt to each propeller’s slipstream. However, this work also shows that RANS-based aerodynamic shape optimization considering propeller-wing interaction yields a negligible improvement compared to optimizing a wing alone without propeller influence. This suggests that regardless of the number of propellers, the added cost of resolving a propeller model is not justified. Further work expanding the design space and more holistically considering the vehicle optimization problem is required to verify this claim for a complete vehicle design optimization problem.
This work shows that increasing the number of propellers on the vehicle can decrease aerodynamic efficiency in cruise flight. However, this study considers only aerodynamic efficiency and does not account for other benefits or disadvantages of distributed propulsion. For example, this work does not include thermal, structural and installation complexities, power plant design and architecture, or potential limitations for control authority of failure modes of limited numbers of propellers. Future work studying the effect of the number of propulsors should also consider variations in required thrust due to varying wing drag, propeller size, propeller location, and propeller rotation direction. Additionally, results obtained for cruise flight should be considered along with transition, climb, and descent mission requirements because the effects of distributed propulsion can be more advantageous in those segments. While the effects of a blown wing with distributed propulsors are likely useful for challenging flight maneuvers such as transition or climb, a wing’s drag in cruise flight suggests that fewer propellers yield better flight performance at the cruise condition.
Acknowledgements
The authors thank Joel Guerrero and Jan Pralits, as well as the rest of the 18th OpenFOAM workshop team, for organizing a productive forum in which to exchange ideas, methodologies, and results built on OpenFOAM. The authors also thank the NASA Glenn Research Center OpenMDAO team for collaborating to establish the optimization framework used in this work. They are grateful to Christopher Silva and the Aeromechanics Branch at NASA Ames Research Center for sharing design data for the NASA tiltwing vehicle design. The computational results of this study were generated using computing resources supplied by the Texas Advanced Computing Center and the Advanced Research Computing Center at the University of Michigan.
Declarations
Ethics approval
Not applicable.
Competing interests
The authors declare no competing interests.
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While the cases ran with 640 GB of RAM, the exact usage throughout the optimization was not recorded.
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Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.
