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# Higher-Order Free-Wake Method for Propeller–Wing Systems

Published Online:https://doi.org/10.2514/1.C034720

## Abstract

A higher-order free-wake (HOFW) method has been developed to enable conceptual design-space explorations of propeller–wing systems. The method uses higher-order vorticity elements to represent the wings and propeller blades as lifting surfaces. The higher-order elements allow for improved force resolution and more intrinsically computationally stable wakes than a comparable vortex-lattice method, while retaining the relative ease of geometric representation inherent to such methods. The propeller and wing surfaces and wakes are modeled within the same flowfield, thus accounting for mutual interaction without the need for empirical models. Time-averaged results found using the HOFW method are compared with experimental propeller, proprotor, and propeller–wing system data, along with two semi-empirical methods. The results show that the method is well suited for performance prediction of lightly loaded propellers/proprotors and propeller–wing systems and can successfully predict design trends. In addition, the time required to define a geometry and solve for the flowfield with the HOFW method as compared to that required with a computational fluid dynamics method make it particularly useful for design-space exploration. These strengths were highlighted through a sample design study on a generic distributed propulsion vehicle.

Nomenclature
$A$, $B$, $C$

circulation coefficients

$AR$

aspect ratio of submerged portion of wing; $bss/c$

$b$

wing span

$bss$

propeller slipstream width

$CD$

three-dimensional drag coefficient

$CL$

three-dimensional lift coefficient

$CP$

power coefficient [$P/ρn3D5$ for propellers, $P/ρA(ΩR)3$ for rotors]

$CT$

thrust coefficient [$T/ρn2D4$ for propellers, $T/ρA(ΩR)2$ for rotors]

$c$

chord

$cl$

two-dimensional lift coefficient

$D$

propeller diameter

$F¯ind$

force due to induced velocities on an surface distributed vorticity element

$Fk¯$

force due to kinematic velocities on an surface distributed vorticity element

$F¯total$

total force on an surface distributed vorticity element

$J$

propeller advance ratio; $V∞/nD$

$m$

number of lifting lines

$n$

propeller rotational speed (revolutions per second), number of spanwise panels

$R$

$r$

$S$

wing area

$s^$

unit vector along the leading-edge bound vortex of a distributed vorticity element

$T/A$

$V∞$

freestream velocity

$w$

axial induced velocity

$y$

spanwise location

$yp$

spanwise location of the propeller

$α$

angle of attack

$αt$

angle of the propeller thrust line with respect to the freestream velocity

$β$

$Γ$

circulation

$γ$

vorticity

$η$

spanwise coordinate in the distributed vorticity element reference frame, propulsive efficiency

$θ$

collective pitch angle of the propeller

$λ$

lift factor for semi-empirical analysis

$μ$

rotor advance ratio; $V∞/ΩR$

$ξ$

streamwise coordinate in the distributed vorticity element reference frame

$σ$

rotor solidity

$ω$, $Ω$

angular velocity of propeller, proprotor

## I. Introduction

Afast and accurate method for performance prediction of propeller–wing systems is desirable for many design applications. In the rotorcraft community, there is interest in continued improvement in tiltrotor cruise efficiency for both military and civil applications [1,2]. There is also a renewed interest in tilt-wing technology, as indicated by NASA’s GL-10 vertical takeoff and landing unmanned aerial vehicle (UAV) [3]. For subsonic transport aircraft, it has been proposed that significant fuel savings can be achieved through high-efficiency open-rotor concepts and advanced turboprops [4]. The use of multiple small-scale electric motors powering tractor propellers distributed across the span of a fixed-wing aircraft is being investigated as a means for drag reduction [5]. Another potential application can be found in the rapidly expanding field of UAVs, which are often propeller-driven. Other aircraft of interest within this field are high-altitude long-endurance UAV, such as the AeroVironment Global Observer and the DARPA-sponsored Vulture [6], both of which feature propellers distributed along flexible, high-aspect-ratio wings.

Traditionally, propellers and wings are designed independently during the conceptual design phase using low-computational cost empirical or simple analytical methods to account for installation effects [7]. Thus, interactions between the two components are not well modeled early in the design process, even though they have been proven to be influential to the performance of each component [1,8]. If the interactions are considered, modeling is typically delayed until later stages in the design process when much of the design is frozen and design freedom is limited. In this later phase of design, more computationally expensive tools can be used to analyze the interaction and better predict performance, but there is minimal opportunity to modify the design to achieve a more beneficial interaction.

For conventional propeller-driven aircraft, this process is arguably sufficient because the design space lies within the range of empirical data and because the interactions are fairly limited in magnitude. For many modern design geometries, such as those noted, this is not the case. In these cases, the design is heavily influenced by interactions between the propeller(s) or proprotor (in the case of a tiltrotor) and wing, and exclusion of these interactions early in the design process impedes the designer’s ability to thoroughly explore the design space. Thus, to maximize design space exploration, a low-computational-cost model capable of capturing all nonnegligible effects resulting from propeller–wing interaction would be beneficial. Such a model must be able to account for full mutual interaction, relevant viscous effects, and wake relaxation to ensure that all significant effects are captured.

Presently, there are several methods capable of modeling propeller–wing interaction that span a range of complexity and accuracy. In general, most have limitations with respect to the conceptual design of the vehicles of interest noted. McCormick [9], for example, recommends the semi-empirical methods of Smelt and Davies [10] and Kuhn [11]. These methods predict several important trends at low computational cost and account for frequently neglected effects, such as viscosity and separation, empirically. Unfortunately, they are not necessarily appropriate for predictions outside of the limited set of experimental data on which they are based. In addition, neither method accounts for the influence of the wing on the propeller or the influence of the propeller geometry.

Methods that rely on physics-based modeling to account for the interaction are more widely applicable, even if they are simplistic in their approach. McVeigh et al. [12], for example, used a converged blade-element momentum theory analysis of the propeller to determine induced velocities (both axial and tangential) and then applied the calculated velocities to a modified lifting-line model. A more recent extension of McVeigh et al.’s approach is that of Stone [13], in which the same blade-element momentum approach for the propeller is combined with a fixed-wake panel method. In these approaches, mutual interaction between the propeller and wing is neglected.

Mutual propeller–wing interaction can be taken into account with fixed-wake potential flow models. In the models of Witkowski et al. [14] and Veldhuis [8], vortex-lattice methods (VLMs) are used to model both the propeller and the wing. Witkowski et al. [14] models the two systems within the same potential field and accounts for viscous effects on the wing with strip theory. Veldhuis [8] iterates between a blade-element momentum solution for the propeller and a vortex-lattice solution for the wing. In this formulation, the induced velocities due to one component are calculated and then imposed on the other. In addition, viscous effects on both the wing and the propeller are accounted for via strip theory.

Although VLMs have been used successfully in some cases, they are limited in accuracy and numerical stability and tend to omit relaxed-wake effects. Specifically, VLMs suffer from numerical discretization errors that, even with a large number of panels, lead to limited accuracy for the prediction of induced drag [15]. In addition, any method that uses discrete vortex filaments to model the upstream surface’s wake, including VLMs, must account for the potential numerical interaction that occurs when vortex filaments, whose induced velocities approach infinite values near their centers, are close to the control points that dictate local circulation on a downstream surface [16]. Finally, some of the causes of the interaction between propellers and wings are a result of deformation of the wake of the propeller due to the wing and the wake of the wing due to the propeller. This includes the deformation of the slipstream due to its impingement on a trailing wing. These types of effects will not be taken into account without wake relaxation.

Relaxed-wake methods such as CAMRAD II [17,18], VSAero [19], and the work of Marretta et al. [20,21] allow for the deformation of the wake via the advection of control points by local induced velocities. Once again, because of the singular nature of vortex filaments, this approach is prone to numerical instabilities [16]. Such conditions occur when the control points in the wake encounter singular velocities when in close proximity with vortex filaments. Relaxed-wake methods mitigate these issues by replacing the singularity at the center of the vortex filament with a finite-core model. The drawback of such an approach is that finite-core models are empirical or semi-empirical in nature and can result in lift and induced-drag predictions that depend on the core model and any cutoff distances that are applied.

Although potential flow methods can account for viscous effects with strip theory, they essentially model the flow as inviscid and irrotational. Removal of the irrotational assumption in propeller–wing modeling requires the use of the Euler equations, as is the approach taken by Dang [22], Whitfield and Jameson [23], and Thom and Duraisamy [24]. Directly including viscous effects requires use of the Navier–Stokes equations (in either a complete or reduced form), which is the approach taken by Roosenboom et al. [25] and Gomariz-Sancha et al. [26]. Although these methods have the potential to capture more of the physics of the propeller–wing interaction, they have limitations in the context of design because they require significant user time for mesh generation and postprocessing [27] as well as additional computational resources to acquire a converged flow solution. For example, in a study by Roosenboom et al. [25], an unsteady Reynolds-averaged Navier–Stokes simulation was used to analyze the slipstream of an eight-bladed propeller impinging upon a wing with remarkable accuracy. To acquire these results, however, required a mesh that used 23 million nodes on the wing and 36 million nodes on the propeller. Even with current high-performance computing technology and state-of-the-art mesh generation, this approach is not well suited for use in conceptual design.

A reduction in computational cost from a full Navier–Stokes analysis can be found in the form of viscous vortex particle methods, such as that of He and Zhao [28]. In these methods, a Lagrangian form of the Navier–Stokes equations is solved for vortex particles within a flowfield. It is thus not necessary to solve for the full flowfield, which reduces the overall computations required and allows for a grid-free approach to wake modeling. Still, the method involves both empirical modeling parameters and coupling with an additional method for surface modeling (such as lifting line or computational fluid dynamics, CFD), the latter of which retains the shortcomings of each method individually.

Because of its accuracy, computational speed, and relaxed-wake capabilities resulting from inherent robustness to singularity issues, the higher-order free-wake (HOFW) method of Bramesfeld and Maughmer [29] offers potential for modeling propeller–wing interaction for conceptual design. Hence, the method has been adapted for this purpose. The current study provides a brief background on the fundamental components of the HOFW method and its extension to propeller–wing systems. The resulting approach is then used to predict the performance of a propeller, proprotor, and several propeller–wing systems, and the results are compared with experimental results and/or other analysis methods. Finally, an exploratory study is provided for evidence of the utility of the method for design-space exploration. This application study investigates the influence of the vertical location of the propellers on the system performance of a generic distributed propulsion vehicle.

## II. Higher-Order Free-Wake Method

The method introduced in this study uses the HOFW method for prediction of lift and vortex-induced drag, augmented with look-up tables to account for profile drag. This approach is an incompressible, higher-order, relaxed-wake, potential flow formulation solved with Neumann boundary conditions. For further details including discussion of the governing equations of the method, the reader is referred to Cole [30]. Its advantages include increased computational speed and numerical stability as well as a realistic prediction of the velocity field. The resulting model, most importantly, provides accurate aerodynamic load predictions for the configurations of interest. In this section, the HOFW method is described in terms of the approach taken for the representation of the lifting surfaces and wakes and the calculation of aerodynamic forces. Within this description, comparisons are made with traditional methods, both to demonstrate differences between the models and to substantiate the relative advantages of the HOFW method.

According to Katz and Plotkin [31], higher-order potential flow solutions are built with elements that consist of “higher-order” (i.e., greater than zeroth-order) distributions of singularity strengths to represent lifting surfaces and their wakes. In this context, higher-order refers to the order of polynomial used to represent the singularity distribution on each panel. The order of accuracy of the overall scheme was not directly evaluated. The HOFW method in particular uses panels with distributed vorticity, referred to as distributed vorticity elements (DVEs). These elements, as depicted in Fig. 1, consist of leading- and trailing-edge vortices (indicated by $Γl.e.$ and $Γt.e.$, respectively) that have quadratic circulation distributions in the spanwise ($η$) direction. The leading- and trailing-edge vortices are connected in the streamwise direction by a vortex sheet having a linearly changing distribution of vorticity strength (indicated by $γ$) in the spanwise ($η$) direction.

A general outline of the method as applied to propeller–wing systems is provided in Fig. 2 in the form of a flow diagram. The steps shown are divided into two main components: initialization and time-step loop. During the initialization, the geometry and operating conditions are prescribed in the global reference frame, thereby defining the body-fixed and DVE reference frames. From this location, the time-step loop is entered, during which the propeller–wing system is moved in the global coordinate frame according to the defined operating conditions. Numerically, the bulk of the solution process involves the determination of the circulation distribution over each surface distributed vorticity element (SDVE) at each time step. This can be formulated as

$[A]{x}={B}$(1)
where $[A]$ is the influence matrix, which is a function of the geometry of the set of DVEs representing the lifting surfaces; ${x}$ is the vector containing the unknown strengths of the surface DVEs; and ${B}$ is the resultant vector in which the boundary conditions are applied. The strength of each surface DVE is defined by the coefficients to the circulation distribution ($A$, $B$, and $C$), as described in Fig. 1.

The lifting surfaces themselves are represented with surface DVEs (SDVEs). An example of the representation of a simple planar wing with a single row of SDVEs is shown in Fig. 3. Spanwise rows of these elements form lifting lines with bound circulations whose strength changes as second-order splines in the spanwise direction. Neumann (flow tangency) boundary conditions are enforced at the control points, and transitional conditions for circulation and vorticity between neighboring DVEs yield the solution for the strength distribution of the bound circulation along each lifting line.

The wake of each lifting surface is developed using time stepping. Such an approach permits time-dependent analysis and tends to provide improved computational efficiency over an equivalent spatial relaxation method [31]. During each time step, the lifting surface is advanced a spatial increment, and a row of DVEs is passed into the wake, bridging the gap between the wing trailing edge and the wake elements of the previous time steps. The wake is assumed to be quasi-steady, which is to say that trailing vorticity shed from the lifting surface at each time step is propagated throughout the wake. As a result, the leading- and trailing-edge filaments on the wake DVEs are omitted from the calculation because they are equal in magnitude and opposite in direction. The subsequent wake representation is a continuous vortex sheet with linearly changing vorticity distribution in the spanwise direction and, as a result, is in effect free of singularities [29,32].

The continuous formulation of the wake vorticity results in a smoother and more realistic velocity field than is the case with lower-order methods, such as vortex-filament methods that are commonly used to represent lifting surfaces and their wakes. As evidence of this statement, the computed downwash velocities induced by two representations of the wake coming of a simple rectangular wing ($AR = 8$) at a moderate angle of attack of ($α=5 deg$) are depicted in Fig. 4. The fixed-wake DVE representation clearly has finite velocities, whereas the vortex-lattice model exhibits large velocities that are singular near the location of the eight vortex filaments (four per half-span). In the region above and below the plane of the wing, the effects of the vortex filaments are still present.

The velocities induced within the flowfield directly influence the numerical stability of the solution. Any unmodified infinite velocity within the flowfield has the potential to interact with a control point and must therefore be addressed. For example, the infinite velocities induced by vortex filaments are commonly rectified with solid core models, as previously discussed. Even with this modification, the induced velocities peak in proximity to each filament. In comparison, the induced velocities in the DVE wake are finite almost everywhere, and thus the approach is more numerically stable than lower-order methods. These properties also make the method suitable for modeling highly interactive systems, such as propeller–wing systems.

The inherent numerical stability and realistic flowfield representation of the HOFW method enables the implementation of wake relaxation at each time step to determine the force-free wake geometry. The relaxation process is governed by the vorticity transport equation as applied to potential flow. Accordingly, the motion of a control point on a wake element is equivalent to that of a coincident Lagrangian fluid particle. Numerically, this process is conducted using an explicit Euler approach, which is first-order-accurate in time. The resulting location of the control points for a given DVE define a new DVE geometry, and the circulation coefficients must be adjusted to maintain a constant value of integrated circulation as required by Kelvin’s theorem of vorticity [33]. The resulting wake is consistent with what would be expected based on experimental wake visualization techniques. For a fixed-wing, for example, the essentially continuous vortex sheet (discretized using DVEs) rolls up at the tips in a manner consistent with the physical system represented [29,32].

Once the flowfield solution is determined, the forces on each SDVE are calculated in the near field using the Kutta–Joukowski theorem. Lift is computed along the lifting lines of the bound circulation considering kinematic velocity of the element and induced velocities. In the case of analysis of a fixed wing, the kinematic velocity is equal to and opposite of the freestream velocity. In the case of a rotating surface, such as a propeller or wind turbine, the kinematic velocity must also include the rotational velocity. Induced drag is determined along the trailing edge of a lifting surface using the kinematic-velocity component of the cross product of the circulation shed into the wake at this location and the wake-induced velocities.

The implementation of the Kutta–Joukowski approach within the method is based on the work of Horstmann [15] but is described here for clarity. The total force due to the lift of each SDVE is the combination of the force calculated with the kinematic velocity and the force calculated with the induced velocity:

$F¯total=F¯k+F¯ind$(2)

These forces are then resolved at each time step into the appropriate coordinate frames to determine the lift and drag (as defined relative to the true freestream velocity) for fixed-wing surfaces and thrust and torque for rotating surfaces. The kinematic lift force is calculated as

$F¯k=ρ∑i=1m∑j=1n[∫−ηi,jηi,j|V¯k×si,j^|(Γi,j−Γi−1,j) dη(V¯k×si,j^|V¯k×si,j^|)]$(3)
where $m$ is the number of lifting lines, $n$ is the number of spanwise panels, $η$ is the half span of the DVE, and $s^$ is the unit vector along the leading-edge bound vortex of the DVE. The forces due to induced velocities are calculated similarly using the local induced velocity instead of the kinematic velocity:
$F¯ind=ρ∑i=1m∑j=1n[∫−ηi,jηi,j|V¯ind×si,j^|(Γi,j−Γi−1,j) dη(V¯ind×si,j^|V¯ind×si,j^|)⋅(V¯k×si,j^|V¯k×si,j^|)]$(4)

The induced velocities necessary for the calculation of induced forces can be calculated at any location within the flowfield based on the strengths of all singularities within the solution and their proximity and orientation relative to the location of interest. This process requires several calculations per DVE, such that as the number of DVEs increases, the number of calculations increases as well. As a result, it can become unwieldy to calculate the velocity at many locations on each DVE. As a compromise between computational efficiency and solution accuracy, the induced velocities are calculated at three spanwise locations along the leading edge of each DVE, and the resulting induced lift is integrated using Simpson’s rule as recommended by Horstmann [15].

A similar process is taken along the trailing edge for the calculation of vortex-induced drag with wake-induced velocities, as is more thoroughly described in Cole [30]. Although the calculation of vortex-induced drag at the trailing edge using the Kutta–Joukowski theorem can only be rigorously proven through Trefftz plane analysis if the wake is fixed and drag-free, both qualitative arguments and comparisons with other methods provide support for the approach with relaxed wakes. Qualitatively, Eppler and Schmid-Göller [34] and Eppler [35] reasoned that the wake shape primarily influences the circulation distribution. Thus, a fully developed wake results in an accurate circulation and lift distribution. All of the wake vorticity is inserted at the start of the wake, and thus with the correct circulation distribution, the vortex-induced drag can be calculated at the trailing edge.

Quantitatively, this approach is supported through comparison with other methods and experimental results [29,32,34,3639]. When used to analyze several planforms, the method predicts span efficiencies within 1% of those calculated with a higher-order panel method using a large panel density. Bramesfeld and Maughmer [29] compared this approach to other methods of induced drag prediction for the HOFW method specifically. They found that when a fixed, drag-free wake is implemented, the span efficiencies of various elliptical planforms calculated using the Kutta–Joukowski theorem at the trailing edge are virtually indistinguishable from Trefftz-plane results. They also compared the span efficiencies calculated with the HOFW method and trailing-edge induced-drag calculation with other theoretical approaches found in the literature and again found excellent agreement in nearly all cases. Schirra et al. [38,39] compared the span efficiencies of highly nonplanar configurations, such as of biplanes and box wings, using predictions based on the DVE-based approach and other theoretical methods, including those of an Euler solver. With the exception of highly staggered box wings, the results compare favorably.

Profile drag is determined using a strip-theory approach along with a table look-up routine. In this process, each lifting surface is divided into chordwise strips that are presumed to act as two-dimensional airfoils. The lift coefficient for each strip is determined based on local conditions. Tabulated airfoil data are then interpolated based on the section lift coefficient and Reynolds number to determine a profile drag coefficient. The tabulated airfoil data can be based on theoretical predictions (typically produced using XFOIL) or experimental results and include the profile drag coefficients at a specified Mach number over a set of Reynolds numbers and lift coefficients. This approach does not account for any boundary-layer effects beyond what is accounted for within the airfoil data and is valid until major flow separation occurs, for example near or poststall.

The HOFW method with the profile drag implementation has been used for several applications including formation-flight aerodynamics [40], design of small and micro aerial vehicles [41], design of sailplane winglets [42], and design and analysis of wind turbines [43,44]. Its numerical speed, stability, and accuracy make it well suited for conceptual design and analysis.

## III. Adaptation of the Higher-Order Free-Wake Method for Propeller–Wing Systems

The aerodynamic modeling of a propeller–wing system presents two main challenges that have not been previously addressed by the HOFW method. First, the geometry and kinematics of each propeller blade must be defined along with the wing to create a physically consistent model of the flowfield at each time step. Second, time dependencies within the flowfield must be taken into account, both in terms of the approach to the flowfield prediction and in the calculation of forces. Both of these steps are addressed within the flow diagram provided in Fig. 2, but further details and justifications are provided in this section.

### A. Development of the Flow Solution

To model a propeller–wing system, the geometry and kinematics of the propellers are defined within the constructs of the original HOFW method as described in the previous section. Thus, both the wing and propeller blades translate within a fixed global reference frame as dictated by a defined freestream velocity, angle of attack, and sideslip angle. The propeller additionally rotates at a rate dictated by the user-defined advance ratio. An example propeller–wing system is shown in Fig. 5. In this example, roll-up of both the propeller and wing wake are clearly visible, along with interaction between the propeller and wing wake. It is also noteworthy that the propeller wake is deformed due to its impingement upon the wing as control points above and below the wing are advected with local velocities. For further details of this process along with an extended discussion of propeller-wake impingement on a trailing wing, the reader is referred to Cole [30].

For performance prediction, time-averaged estimates of lift, drag, thrust, and torque are desired. Despite this, it is in the time-accurate solution that interactions between the propeller and wing take place. For example, the lift at a location on the wing at a moment in time may be increased due to its proximity to a propeller blade. This increase in lift will result in a change in the velocities induced by the wing on the propeller wake. The propeller wake will then deform due to the change in local velocity, which may in turn change the performance of the propeller. Without the ability to calculate this time-accurate interaction, it is not possible to predict true time-averaged flowfield or, transitively, the time-averaged performance.

The HOFW model accounts for these sources of mutual interaction through flow tangency boundary conditions and development of the force-free wake. To determine the circulation distribution on all lifting surfaces (including both the propeller and wing), flow tangency is enforced at the control point of each surface DVE while taking into account the velocity induced by all lifting surfaces and wakes. The inherent stability of the DVE wake, as discussed in the previous section, allows for this interaction without the need for solid cores or artificial damping of the induced velocity field. Likewise, control points in the wake of both the propeller and wing are advected with the velocities induced by all lifting surfaces and wakes to develop a force-free wake. It is worth noting also that the wake relaxation process is valid for both steady and time-dependent systems due to Kelvin’s theorem of vorticity, which holds in both steady and time-dependent flows [33].

### B. Force Calculation Approach

As in the original HOFW method, lift and drag are calculated locally on each lifting surface using the Kutta–Joukowski theorem, and viscous effects are taken into account through the addition of profile drag. These forces are then resolved into freestream lift and drag on the fixed-wing surfaces and thrust and torque on the rotating surfaces. To account for a time-dependent flowfield, the integrated forces are averaged over one full propeller revolution.

The use of the Kutta–Joukowski theorem for the calculation of lift on a time-accurate basis has been implemented by Drela [45] and in similar form by Katz and Plotkin [31], although it is described in the latter reference as a pressure integration. The specific approach taken within the HOFW method may be categorized as quasi-steady [46]. At each time step, the bound circulation on all lifting surfaces is adjusted to enforce flow tangency based on the velocities induced by all surfaces and wakes, but the influence of shed vorticity due to changes in circulation on the lifting surface and noncirculatory effects are neglected. Because the interaction between the propeller and wing is periodic in nature, the inclusion of shed vorticity and noncirculatory effects can be expected to change the amplitude and phase of oscillations but not the mean value. Thus, their influence is assumed to be negligible for the purpose of time-averaged predictions, as confirmed through comparison of performance predictions made with the HOFW method to semi-empirical methods and experimental results as provided in later sections. Further investigation of the influence of shed vorticity and noncirculatory effects is provided in Cole et al. [47].

Use of the Kutta–Joukowski theorem to calculate vortex-induced drag raises several questions in the setting of propeller–wing systems. The propeller adds energy to the flow that must be considered. In addition, the periodic unsteady nature of the flowfield about the wing must be addressed. Finally, in light of these concerns, a reexamination of the influence of wake relaxation is warranted.

A Trefftz-plane analysis of a propeller–wing system featuring a time-averaged wake for the propeller and a fixed, drag-free wake for the wing is used to address the energy added to the flow by the propeller. For this to be done, the validity of time-averaging for the purpose of control volume analysis must first be addressed. The interaction between the propeller and the wing can be assumed to be periodic-unsteady for control volume analysis according to Drela [48]. As such, and assuming that there is no “large-scale unsteadiness” as is present in flapping-wing systems, any terms neglected due to time-averaging are expected to be small. Thus, it is reasonable to conduct a Trefftz-plane analysis on a propeller–wing system using the time-averaged flowfield.

Through such a Trefftz-plane analysis, it can be shown that, after thrust and torque are taken into account, the losses due to interaction between the propeller and the wing are equivalent to those that would be calculated with the Kutta–Joukowski theorem applied at the trailing edge of the wing [30]. In this case, the downwash velocity includes both the downwash induced by the wake of the wing (as in traditional analysis) and the velocities induced normal to the wing surface by the propeller. It is important to note that these losses are in addition to the swirl losses due to the rotation of the propeller wake that degrade the propeller performance in isolation. This is to say that, in agreement with the literature [8,49], the interference between the propeller and wing results in losses beyond what each system experiences in isolation.

In the Trefftz-plane discussion, the interference losses are combined into a single term calculated for the wing. In reality, and in the HOFW method, interference affects the performance of both the propeller and the wing due to a combination of factors. Physically, the wing induces velocities (vertically and horizontally) upstream of its location. These velocities change the propeller inflow conditions, thus affecting the thrust distribution and, subsequently, the propeller wake strength and the propeller power. The modified propeller wake then influences the wing lift distribution changing the induced drag properties of the wing. If the two systems are separated in the streamwise direction such that propeller is well upstream of the wing, the interference losses are experienced entirely by the wing, as is the case discussed previously. Alternatively, if the wing is well upstream of the propeller, the losses manifest in terms of the thrust of the downstream propeller. This is in agreement with the results of Kroo [49], who addressed the influence of the streamwise position of the elements, given a fixed circulation distribution. When the two systems are close enough to interact, the fixed-circulation assumption is violated, and the interference can potentially manifest in both locations.

Within the HOFW method, the interference losses for each component are taken into account through two mechanisms. First, the circulation distribution on each lifting surface is determined through the application of boundary conditions that include the influence of all other lifting surfaces and wakes. Thus, there is no “fixed-circulation” assumption in the HOFW method, and mutual interaction is intrinsically taken into account. Second, the induced drag for each component is calculated locally along the trailing edge of the lifting surfaces with velocities induced by all wake elements within the flowfield. Accordingly, all relevant local phenomena are incorporated into the calculation.

A key assumption applied in the Trefftz-plane analysis was that of fixed, drag-free wakes. Evidence suggests that the difference between the induced drag as calculated with a fixed wake and a relaxed wake for the same circulation distribution is typically small. For example, Smith [50] showed that, as long as deflection angles and curvature in the wake are small, a drag-free wake may be substituted for a relaxed wake at the trailing edge of the lifting surface without inducing an error greater than 1.5%. For the application at hand, it is noteworthy that the wing wake experiences additional deflection due to the propeller slipstream. The magnitude of the tangential induced velocity in a propeller wake depends on the propeller geometry and operating conditions, but for lightly loaded conditions ($T/A<10 lb/ft2$), it can be assumed to be less than 5% of the freestream velocity. This would result in an additional deflection angle of approximately 3 deg, which would presumably not have a significant effect. At higher disk loadings, this assumption should certainly be revisited.

A proof showing that the calculation of induced drag on a lifting surface with a relaxed wake is equivalent to that on a lifting surface with a fixed, drag-free wake with the same circulation distribution would be ideal because it would show that the results of the Trefftz plane analysis apply directly to a relaxed-wake system. To overcome the lack of a rigorous proof on this topic, the following logical argument is used. The Kutta–Joukowski theorem applied at the trailing edge can be confirmed through Trefftz plane analysis if and only if the lifting surface is modeled with a fixed, drag-free wake. Despite lacking such a confirmation, evidence suggests that this approach also works for a relaxed wake [29,32,34,3639]. Thus, the properties relating to the induced drag of a relaxed wake are implied to be consistent with those of a fixed wake. If this is the case and the Kutta–Joukowski approach is valid for a fixed, drag-free wake model of a propeller–wing system, then it is also valid for a relaxed wake model.

A few caveats to these arguments must be mentioned. First, in order for this argument to be true, the lifting surface modeled with a relaxed wake must have the same circulation distribution as that modeled with a fixed wake. It is demonstrably false that the induced drag of a system in which the flow tangency has been enforced while incorporating wake relaxation effects is necessarily the same as one in which flow tangency has been enforced using a fixed, drag-free wake. Second, this argument does not prove that a fixed, drag-free wake is equivalent to a relaxed wake. It implies only that evidence suggests that they are reasonably consistent for the purposes of calculation of induced drag. This conclusion is supported by the reasoning of Eppler and Schmid-Göller [34] in that, in both models, all of the energy of the wake due to the lifting surface is present at the trailing-edge location.

Although it was addressed in the Trefftz plane analysis, a revisiting of energy addition due to the propeller may be warranted in the context of relaxed wakes. This energy is accounted for in the model through velocities induced by the rotating lifting surfaces and their wakes. In this respect, the propeller–wing system is equivalent to an interacting multiple wing-wake system as has been specifically addressed in the work of Bramesfeld and Maughmer [40] for relaxed wakes.

## IV. Comparison with Other Analysis Methods and Experimental Results

Having described and justified the formulation of the HOFW method, it is necessary to ensure that it is capable of prediction of the performance of real world propeller–wing systems. The objective of this section is to provide evidence of this through comparison with the results of other analytical methods and experiments. Because the method has been previously examined for fixed-wing analysis, the focus of this section is propellers/proprotors and propeller–wing systems.

### A. Propeller/Proprotor Performance

To ensure that the method is able to accurately predict the performance of propulsion systems, experimental data for one propeller and one proprotor were compared with the performance estimates found using the HOFW method. The propeller considered is a 10-ft-diam propeller tested by NACA and designated P5868-9 [51]. The HOFW-predicted power coefficients $CP$, thrust coefficients $CT$, and propulsive efficiencies $η$ as a function of advance ratio for three blade pitch angle settings are compared with the experimental results in Figs. 68, respectively.

For lightly loaded conditions (moderate to high advance ratios and low to moderate pitch angles), the predictions match the experiment well. At 15 deg pitch, the average error in efficiency across the full range of advance ratios is approximately 2%, increasing to 4% at 25 deg pitch and 10% at 35 deg pitch. The largest errors occur at low advance ratios. In this region, rotational effects have a strong influence on the two-dimensional section characteristics, resulting in large delays in stall onset [52]. The actual stall model applied omits these effects and is intended only for the identification of two-dimensional stall onset rather than poststall performance prediction. Another source of error lies in the implicit assumption that the section lift-curve slope is a consistent $2π$ per radian, when in reality the lift-curve slope depends on the airfoil and decreases significantly as the airfoil approaches stall. As the propeller pitch angle increases, it is possible that both of these errors come into effect, resulting in the increased error for increased pitch. It is additionally possible that geometric discrepancies between the test propeller and modeled propeller are a contributing factor. Still, without accounting for rotational effects and improving the approach to modeling stall, the current method is best suited for lightly loaded propellers at moderate to high advance ratios.

The JVX proprotor [5355] is frequently referred to as a 0.656-scale V-22 proprotor, although in reality there are slight differences between the two designs. Regardless, it provides a good basis for validation of the method for proprotors. As such, it was investigated with the HOFW method over a sweep of collectives at two advance ratios, resulting in the performance predictions shown in Fig. 9. Also shown in Fig. 9 is a performance prediction using CAMRAD II and the original test data [55]. In Fig. 9, the advance ratio $μ$ and coefficients are defined according to the rotorcraft conventions, and $σ$ refers to the rotor solidity (ratio of blade area to disk area). The performance predicted by the HOFW method closely matches the test data and the predictions using CAMRAD II both in shape and magnitude. This is to be expected based on the P5868-9 results because the proprotor is lightly loaded in the cruise condition.

### B. Series of Conventional Propeller–Wing Systems

Although the primary intended application for the HOFW method is for the design of nonconventional propeller–wing systems, a first step is to compare with conventional semi-empirical methods for prediction of propeller–wing interaction. This comparison is useful because these semi-empirical methods take into account all relevant effects for conventional systems as they are based at least in part on an empirical case base. Thus, the impact of assumptions made within the HOFW method can be assessed in this scenario with a high degree of confidence. In addition, although comparisons of the HOFW method and another analysis method (such as CFD) or an experiment are subject to the potential shortcomings of the secondary method or experiment, comparison with well vetted semi-empirical methods over a range of designs provides broader support. Accordingly, a series of conventional propeller–wing systems were analyzed with two semi-empirical methods and the HOFW method. The series of propeller–wing systems consist of wings of constant span ($b=10 ft$) and varying chords at an angle of attack of 5 deg in the wake of two propellers designed using QMIL [56], for lightly loaded conditions ($CT≈0.04$ and $CT≈0.17$). The two semi-empirical methods are those of Smelt and Davies [10] and the jet-flap approach recommended by McCormick [9]. The wings were each analyzed in isolation as well to determine quantities necessary for use in the semi-empirical methods.

According to Smelt and Davies [10], the change in the lift coefficient of a wing due to a propeller slipstream can be estimated as

$ΔCL=clDcSwV∞λ$(5)
where $D$ is the diameter of the propeller, $c$ is the chord of the wing, $w$ is the axial induced velocity, $S$ is the wing area, and $V∞$ is the freestream velocity. The lift coefficient $cl$ is the average two-dimensional lift coefficient of the region of the isolated wing that would be engulfed in the propeller slipstream. The lift factor $λ$ lies between 1 and 2, with limiting cases derived by assuming that the wing chord is much larger than the slipstream diameter (for $λ=1$) and much smaller than the slipstream diameter (for $λ=2$). In the latter case, it was assumed that the induced effects due to the added vorticity on the wing are negligible, which is akin to assuming that wing can be practically treated as two-dimensional. The lower bound is explained as the case in which shed vorticity is dominant, although the explanation is qualitative in nature and does not directly justify the quantitative value [57]. This approach is restricted to use with clean (nonflapped) wings and lightly loaded, upstream propellers.

McCormick [9] recommends treating the section of the wing submerged in the propeller slipstream with a semi-empirical method developed for analysis of jet flaps. This process requires an estimation of deflection of the slipstream due to the wing and uses this estimation to account for the change in momentum of the deflected slipstream. For more information on this approach, the reader is referred to McCormick [9].

A comparison of the change in the three-dimensional lift coefficient due to the propeller versus the aspect ratio of the submerged portion of the wing (slipstream width/wing chord) as predicted with the HOFW method and the two semi-empirical methods is provided in Fig. 10. The three methods seem to agree on the general magnitude of the increase in $CL$ and, at least for the higher thrust coefficient, the trends as well. Without experimental data with which to compare, it is difficult to determine which of the three models is “best” for this application. If the average of the two semi-empirical methods is taken as a reasonable baseline, the average error in the HOFW prediction of the more lightly loaded case is around 6% and jumps to around 13% for the more heavily loaded case. Overall, the agreement supports the case that the HOFW method is capable of predicting trends for propeller–wing systems.

### C. Nonconventional Propeller–Wing System

The forthcoming comparison is based on the experiments conducted by Dunsby et al. [58]. This case was selected because the geometry is nonconventional and heavily emphasizes propeller–wing interaction. A dimensioned diagram of the model is provided in Fig. 11. The wing section uses an NACA 0015 airfoil and the wing Reynolds number is approximately 1 million in all cases. The four-bladed propeller rotates in the down inboard orientation, and the blades have a simple rectangular planform with a linear twist distribution and NACA 1-series airfoils. The propeller was modeled according to the geometry and operating conditions described in Dunsby et al. [58]. For the case considered, the propeller operated at a rotation frequency of 3000 rpm, and the tunnel speed was approximately $95 ft/s$.

To understand the model accuracy, we performed unsteady Reynolds-averaged Navier–Stokes analyses to the propeller–wing system within the wind tunnel. The CFD was performed in the context of the StarCCM+ CFD code [59]. The model uses the Spalart–Allmaras turbulence model with a sublayer-resolved wall-normal spacing (hence, it has roughly wall spacing of 1.0 or less in terms of wall units) [60]. In handling the relative motion between the propeller and wing system, a sliding interface is used to couple a rotating mesh resolving the propeller to a mesh fixed to the wing/wind-tunnel system. In this sliding interface, cross-mesh interpolation is used to provide a flux-conserving interface. The overall size of the computational mesh was approximately 6 million cells. Note that symmetry was taken advantage of, and a plane of symmetry was assumed along the centerline. The model imposed the measured velocity from the experimental to the inflow, providing a constant inflow boundary condition. At the outflow of the wind tunnel, a constant pressure was prescribed. The walls on the wing and propeller were all treated as no-slip walls. It is noteworthy that several geometric approximations were made, which include neglecting the sting, the nacelle, and the hub of the propeller. These could have been approximated; however, the goal of the CFD was to evaluate the accuracy of the HOFW with respect to the underlying physical assumptions (i.e., inviscid) and give confidence in the approach used for force calculation. Hence, the CFD model was aligned with the HOFW model, which neglects the nacelle and hub. The one difference between the two was that, to better match the experimental thrust values, the propeller pitch was adjusted slightly by approximately 2 deg in the CFD. The overall CFD solution required roughly 1000 time steps to remove transients and an additional 100 time steps for unsteady data. A contour plot of the pressure coefficients [i.e., $cp=(p−p∞)/((1/2)ρV∞2)$] for the zero angle of attack case is shown in Fig. 12. These results indicate the expected flowfield and character. These results are later used to provide an evaluation of the HOTW results with respect to the loading, loading distribution, and drag distribution.

The influence of temporal and spatial resolution on the HOFW method solutions is discussed in Cole [30]. The HOFW model was run with 16 panels per propeller blade and 18 panels per half-span at a time step dictated by 20 time steps per revolution of the propeller. A single lifting line was used on the propeller blade, and 10 lifting lines were used on the wing. This model was run out to 50 time steps to achieve convergence in time-averaged lift, drag, thrust, and torque. The resulting solution is shown in Fig. 13. This computation took approximately 35 min on a single core i5 processor.

The time-averaged integrated lift coefficient as a function of angle of attack is shown in Fig. 14. The CFD best matches the experimental results across the range of angles of attack with an average difference of 0.0235 in lift coefficient. The HOFW method, with an average difference of 0.040 in lift coefficient, is only slightly less accurate when compared with the experiment. Smelt and Davies’s method [10] is fairly close, despite the simplicity of the model, with an average difference of 0.054 in lift coefficient, whereas the jet-flap approach recommended by McCormick [9] is less successful for this geometry (particularly in the high and low angles), with an average difference of 0.213.

To get a more detailed view of what is happening in terms of the lift, a comparison of the lift distribution at $−5$, 0, and 5 deg angle of attack found experimentally is compared with the HOFW and CFD results as shown in Fig. 15. The influence of the propeller seems to be somewhat overpredicted in both HOFW and CFD locally, as is visible outboard at 5 deg and inboard at $-5 deg$ as compared to the experimental results. In addition, higher-frequency content in the CFD lift response near the center of the propeller is likely a combined effect of the lack of nacelle modeling and viscous wake effects near the root of the propeller blades. Still, the matching between the CFD and HOFW is remarkable, given the difference in the complexities of the models. It is also worth noting that differences in integrated lift between the experiment and analysis methods are extremely small for these three angles of attack (see Fig. 14).

The time-averaged integrated drag coefficient as a function of angle of attack is shown in Fig. 16. The error in between the models and experiment is much higher in drag than was seen in the lift response. This error can be attributed to several factors. First, the experimental results are based on force balance measurements where the thrust is subtracted to estimate drag. The thrust is measured with strain gauges within the nacelle, and the authors of the study cite measurement uncertainty of 5% with respect to this measurement. Because the magnitude of the drag is small compared to the thrust, this uncertainty translates into a high uncertainty for drag as well. Second, the CFD and HOFW models do not account for nacelle effects, including the increase in pressure drag at higher angles of attack, blockage due to the nacelle, juncture flow, and differences in flow near the root of the propeller blade due the presence of the nacelle. Finally, the difference between the CFD and HOFW predictions can be attributed at least in part to the fact that the CFD is fully turbulent, whereas the HOFW method accounts for transition through look-up tables.

The trend in drag coefficient is somewhat similar between the two methods, but there is a difference between the models and experiment from $−5$ to $+5 deg$ in that the increase in drag seen in the experiment is skewed to lower angles of attack, whereas the model increase is skewed to higher angles of attack in both cases. This could be a function of the influence of the nacelle or error in the experimental results, but the source of the discrepancy is unclear.

There is an average difference of 0.02 in drag coefficient between the CFD and HOFW method across the angles of attack, where the HOFW method consistently underpredicts relative to the CFD. To investigate this discrepancy further, the spanwise distribution of drag at $−5$, 0, and 5 deg angle of attack is provided in Fig. 17. The trends are remarkably consistent across the span, particularly with regard to the influence of the propeller wake. These results lend credence to the theory that the lack of transition modeling in the CFD is a contributing factor. They also provide confidence in the HOFW method’s ability to correctly predict the influence of the propeller on the wing at a design level.

Both semi-empirical approaches for wing lift require propeller thrust as an input, which presents an additional challenge with respect to use of these methods. In the case of the Dunsby et al. experiment [58], the thrust was determined experimentally (albeit with the caveats mentioned previously) and could thus be implemented directly. If the thrust is not known a priori for a given system under specific operating conditions, it must be estimated or calculated using a different analysis method. The other method may or may not allow for the influence of the wing as well, resulting in the possible need for a third correction. In the HOFW method, the thrust is predicted using the same construct as the lift and drag, and thus changes in operating conditions and the influence of the wing are automatically taken into account.

The thrust predicted by the HOFW method is compared with the experimental results in Fig. 18. The trend in thrust with angle of attack present in the HOFW is in agreement with the experimental results at low angles of attack ($−5$ to $+5 deg$). At the higher angles of attack ($±10 deg$), the experimental thrust increases more rapidly than is predicted by the HOFW analysis method. It is plausible that this departure is caused by not accounting for nacelle effects (including blockage and differences in the flowfield near the root of the propeller blades due to the nacelle) in the analysis method and/or by errors within the experiment. The latter conclusion is anecdotally supported by experimental data from Kuhn and Draper [61] that indicates that an increase in thrust of 20% from the zero alpha condition for a propeller–wing system would require an increase in angle of attack of approximately 30 deg. Finally, the magnitudes of the HOFW predictions are relatively close to the experimental results (within 5%) in the low angle of attack range.

In addition to thrust, the HOFW method is capable of predicting the torque of the propeller to provide a full prediction of the performance of the system. This again is not covered by the semi-empirical methods directly and would require the addition of a secondary model or approach. A comparison of the efficiency of the propeller (which takes into account the torque) as predicted by the HOFW method with experimentally derived values is provided in Fig. 19. Although it appears that some error exists within the experimental data, confidence is provided by the closeness of the magnitude of predicted efficiency of the HOFW across the angles of attack.

Overall, the HOFW method satisfactorily predicts the thrust, torque, and lift of the propeller–wing system examined by Dunsby et al. [58]. Specifically, it is more accurate in lift than either semi-empirical method, and it provides much more detailed information (such as lift and drag distributions) than would be readily available from the empirical methods. Confidence in the approach taken to force predictions within the HOFW method is increased through the favorable comparison with CFD in spanwise lift and drag distributions.

### D. Comparison of Prediction of Design Trends with Experiment

To use the HOFW method for design of propeller–wing systems, it is important to show that, in addition to capturing the performance of a single design, the method is capable of predicting trends with changes in design. As such, performance predictions found using the HOFW method were compared with the experimental results of Veldhuis [8].

Veldhuis [8] conducted a series of experiments to determine the influence of propeller location on downstream wing performance. To do so, the propeller–nacelle assembly was separated from the wing such that each could be moved independently. The specific parameter considered in this study is that of the propeller’s spanwise location at a set wing angle of attack (in this case, 4.2 deg). The propeller axis location was varied from approximately 30% of the half-span to the wing tip with the rotation direction being up-inboard. The axis of rotation of the propeller was kept parallel with the freestream velocity and coincident with the wing quarter-chord location. The study was conducted at a thrust coefficient $CT$ of 0.10. Further details of this experiment can be found in Veldhuis [8].

The change in lift coefficient, drag coefficient, and lift-to-drag ratio $L/D$ from calculated mean values found using the HOFW as compared to those found experimentally by Veldhuis are provided in Fig. 20. There is agreement between the experiment and HOFW model in trend for all three metrics everywhere except when the propeller is aligned with the wing tip. The error at this location is assumed to be a result of slight differences in geometry and propeller operating conditions between the analytical model and the experimental one, in particular the lack of nacelle modeling within the HOFW method. Because the flowfield at the wingtip is particularly influential to the induced drag on the wing, omission of the nacelle wake at this location could result in a significant change in performance. For reference, the mean lift coefficient was overpredicted by the HOFW method relative to the experimental results by 2.8%, and the mean drag coefficient was underpredicted by 12.2%.

## V. Discussion of Computational Time

Having provided evidence of the accuracy of the method, it is necessary to address the computational time of the method to ensure that it is well suited for design studies. There are two factors that influence the start to finish time of the analysis of a design. The first is the time required to generate the geometry itself, and the second is the time required to acquire a solution. In both respects, the HOFW affords some significant advantages when compared with a CFD-type method.

The differences between defining a geometry and flowfield within the HOFW method and a CFD-type method are notable. Within the HOFW method, the lifting surfaces are defined with the quarter-chord location, twist, and chord of each panel end point within the body-fixed coordinate system. Only a single propeller blade need be defined within the input file. This propeller blade is then repeated, rotated, and translated according to the input number of blades per propeller, number of propellers, propeller axis location, and propeller rotation direction (as dictated by the individual advance ratio). Finally, the airfoil for each panel is defined by the specification of a look-up file, which spans a range of Reynolds numbers. In a CFD method, the definition and detailed mesh-generation process are significantly more complex. The full physical geometry must be defined in three-dimensional space, and independent motion of surface requires special considerations. In addition, to ensure a “valid, high-quality mesh” for each case, user attention and interaction is required [27].

An investigation of the spanwise chord distribution on a wing within a propeller–wing system provides an example comparison of these two approaches in the context of design. Using the HOFW method, the only adjustments necessary are in terms of the chord length input. The airfoil Reynolds number range will account for any changes in local Reynolds number, and there is no need to adjust the panel resolution in the method. This design sweep can easily be automated to be run on a computer cluster in parallel or in series. In contrast, each design change in CFD would require a redefinition of the geometry details, i.e., cross section and variations between cross sections. In addition, each configuration would require remeshing and a reexamination of each mesh.

Once the geometry is defined, a flowfield solution must be calculated. It is difficult to compare the time required to solve for the flowfield using the HOFW method to that required with a CFD method because the latter heavily depends on the resolution needed and computational availability. Certainly, in the majority of cases, one could expect a faster solution using the HOFW method than a CFD method primarily because the flowfield is determined by solving for surface boundary conditions only in the HOFW method, rather than solving for the entire flowfield as in CFD. Anecdotally, the very high-resolution CFD conducted on the Dunsby et al. [58] case took 288 processor hours, whereas that conducted using the HOFW method took about 0.6 processor hours, almost three orders of magnitude less.

One technique to further reduce the computational time of the HOFW method is to either “freeze” the wake after a certain streamwise distance from the trailing edge of the wing or to implement a fixed wake for the full system. The first option is akin to implementing a near-wake/far-wake cutoff. Further discussion of this approach and its implementation is provided in Cole [30], where for some cases it was shown to reduce computational time by up to 60% with very little influence on the resulting performance prediction. Fixed-wake analysis reduces the computational time further but is typically only applicable for very lightly loaded propeller cases ($T/A<5$$lb/ft2$) and planar wings [62].

The differences in geometry definition, computational time, and space requirements discussed in this section support the use of the HOFW method for design. In particular, the method is well suited for automated design sweeps that can cover a large region of design space quickly.

## VI. Distributed Propulsion Design Study

A sample design study was conducted on the propellers and wing of a generic distributed propulsion vehicle to provide evidence of the utility of the method. This system features seven two-bladed propellers per half-span grouped in a two–three–two pattern upstream of a simple rectangular wing, as shown in Fig. 21. The propellers rotate down-inboard at a constant advance ratio $J=1.6$ in all design cases, with adjustments in thrust achieved through small changes to propeller pitch. The design parameter investigated is the vertical location of the propeller axes with respect to the quarter-chord location of the wing, with the objective being to maximize the efficiency of the overall system.

To accomplish this objective, it is necessary to first identify an appropriate metric for system efficiency. The typical metrics employed for this purpose are the wing lift-to-drag ratio $L/D$ and the propeller efficiency. The drawback of using $L/D$ for this application is that it does not take into account any changes in propeller thrust or power resulting from a change in wing design. Likewise, an optimization of propeller efficiency does not account for any resulting changes to lift and drag of the wing. Because the propellers and wing interact with one another, these metrics on their own are not sufficient to determine if a design change has improved the overall system.

To take both the propeller and wing efficiency into account, one approach is to consider the trimmed propeller power required [30]. To do so, a trim condition must be defined based only on the propeller and wing performance characteristics. This can be accomplished through the use of two nondimensional coefficients. The first coefficient is that of excess thrust, $Cx$, defined as

$Cx=Tcos(αt)−D(1/2)ρV∞2S$(6)
where $T$ is the thrust, $αt$ is the offset of the thrust line from the freestream velocity, $D$ is the wing drag, and $S$ is the wing area. The excess thrust at a given lift condition is physically representative of the allowable drag of the rest of the aircraft such that the trim condition can be met. The second nondimensional coefficient $CL*$ is that of the lift coefficient augmented with thrust contributions in the lift direction (again due to the offset of the thrust line from the freestream velocity). The trimmed propeller power required can then be identified by interpolation of propeller power predictions to the selected trim conditions. An example of this type of calculation is provided in Fig. 22. In this case, a coarse sweep of propeller pitch angles at a set advance ratio was considered, and two-dimensional linear interpolation was used to find the approximate average propeller power coefficient for a cruise lift coefficient and a selected $Cx$ value.

The predicted averaged propeller-power coefficients found using this process with an assumed $CL*$ of 0.75, $Cx$ of 0.005, and set propeller geometry and advance ratio of 1.6 are provided in Fig. 23. The results indicate that placing the propeller axis below the wing can reduce the propeller power required by about 5% when compared with the above axis scenario. This is an interesting result in that, according to Prandtl [63], placement of the propeller above the wing should increase the wing efficiency, whereas placement of the propeller below the wing should increase in propeller efficiency.

To further explore this possibility and to identify the driving force behind the reduction in propeller power, the results of Fig. 23 were broken down into changes in average propeller thrust, wing drag, and average propeller power, as presented in Fig. 24. It should also be noted that changes in wing lift and angle of attack are negligible across all cases. In all cases, the thrust and drag vary directly, as would be expected because as the drag increases, the thrust must increase to maintain the trim condition. The percent changes in wing drag exceed those of averaged thrust due to the relative magnitude of the dimensional values (recall that thrust is trimmed to exceed wing drag to account for the rest of the drag of the aircraft). If linearity is assumed between the propeller thrust and torque over small changes, the interaction between the propeller and wing systems can be seen in the difference between the percent change in thrust and percent change in power. For example, in all three below-the-wing cases, the percent decrease in propeller power exceeds that of the thrust, indicating favorable interaction between the systems. Alternatively, in the above-the-wing cases, this delta is much smaller, and in the case of $z/R=0.333$, the percent increase in propeller power exceeds the percent increase in thrust, indicating an adverse interaction. These data support Prandtl’s assertion that the propeller is more efficient under the wing for this case. Alternatively, the wing drag generally increases for propeller locations above the wing; thus, counter to Prandtl’s assertion, the wing efficiency (as measured in $L/D$) decreases for propeller locations above the wing in this case. This is likely a combined effect of profile drag and induced drag, both of which are influenced by the change in the spanwise lift distribution resulting from a change in propeller position.

The total computational time for this study was approximately 1100 processor hours, where each case was run on a single processor in slightly over 13 h. This allowed for analysis of seven designs at 12 operating conditions each (four angles of attack and three propeller pitch angles), or 84 cases total. The parameter changes were automated such that the study was run nearly in its entirety without human intervention.

## VII. Conclusions

A higher-order free-wake (HOFW) method for performance prediction of propeller–wing systems has been developed. Through comparison with other methods and experimental data, it has been shown that the method can be used with confidence to predict the performance of a lightly loaded propeller or proprotor and propeller–wing systems. The method matched experimental propeller efficiencies to within 4% for lightly loaded conditions. Increases in lift coefficient due to interaction with a propeller for a series of wings as analyzed with the HOFW method were within 6.5% of the average of those predicted with two semi-empirical methods for a lightly loaded propeller. A comparison of HOFW predictions of lift for a more nonconventional propeller–wing system with experimental results over a range of angles of attack showed an average difference of 0.04 in lift coefficient. For this nonconventional system, predictions in thrust and torque also matched experimental results within 5% over a small angle-of-attack range ($±5 deg$). The method was less successful at predicting the magnitude of drag in comparison with experimental results but was capable of matching trends in drag for variations in design. Along with its accuracy in predicting time-averaged performance, the strengths of the method include its resolution per panel in terms of the circulation distribution, its favorable numerical stability for relaxed-wake modeling, and its quickness in both geometry definition and solving for the flowfield. These strengths were highlighted through a design study on a generic distributed propulsion vehicle.

## Acknowledgments

This research was partially funded by the U.S. Government under agreement W911W6-11-2-0011 and through support by the Pennsylvania State University Vertical Lift Research Center of Excellence. The authors would also like to gratefully acknowledge the technical contributions of James Coder and Robert Kunz from the Pennsylvania State Applied Research Laboratory. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government.

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