1 Introduction

Fig. 1
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Considered propeller wing configurations

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As part of the LuFo project GNOSIS, possible aerodynamic advantages of electrically powered aircraft are investigated. The variable position of lighter and more compact electric motors enables new unconventional propulsion integrations, which raises fundamentally new questions in aerodynamics. In a broader sense, the term distributed propulsion (DP) covers all propulsion systems with more than two units [1], in a narrower sense as used here, it means propellers distributed along the wing span. The idea behind distributing a larger number of smaller propellers over the wing is to exploit the interactions that occur between the propellers and the wing to increase the overall efficiency of the aircraft. In tractor arrangement, a part of the kinetic energy in the propeller slipstreams can be used to increase the aerodynamic efficiency of the wing, before it is lost through dissipation. A part of the axial kinetic energy inside the slipstream generates a lift increase at the wing [2]. In addition, a part of the rotational kinetic energy in the swirl of the propeller slipstream can be beneficially recovered by the wing (swirl recovery). The upwind induced by the propeller turns the resulting force vector forward in the area covered by the side of the ascending propeller blades. In the case of a finite wing and a rotation direction opposite to the wingtip vortices, the effects of the ascending and descending blades do not balance each other out and, as a result, the total induced drag can be reduced. Tractor propellers thereby increase the aerodynamic efficiency of the wing. A special case are wingtip mounted propellers (WTP), where only the side of the ascending blades covers the wing and directly counteracts the wingtip vortex, resulting in the largest reduction of the induced drag [2]. Furthermore, the propellers can also benefit from the interactions. When positioned in the area of reduced inflow velocity due to the presence of the wing, the efficiency of the propellers increases. In addition, when the total propeller disk area is increased in the design of DP, the volume of airflow passing through the propellers increases, which in turn results in a higher propulsive efficiency [3]. Propeller efficiency and aerodynamic efficiency are thus dependent on the configuration of propellers and wing. Different concepts with interacting propulsion systems such as DP and WTP are currently being investigated. DP configurations are subject of research by the DLR [4, 5], at the TU-Delft [6], the NASA with the projects X57 [7] and Pegasus [8], as well as the Onera with the Ampere project [9]. The positive effects on WTP were already demonstrated in wind tunnel tests in the 1960–80s [10,11,12,13,14]. Design studies on WTP configurations have recently been carried out by Stokkermans [6] and by Blaesser [8]. However, a general aerodynamic comparison of the concepts under the same boundary conditions with regard to their energy efficiency has not yet been carried out. The present paper fills this research gap by numerically analyzing three principally different configuration concepts (full DP, partial DP and WTP) and comparing them with each other and with a conventional reference configuration, see Fig. 1.

2 Reference configuration and methodology

In the first part of the GNOSIS project, a 19-seater commuter aircraft is considered to evaluate the influence of electric propulsion. For this purpose, a twin engine Beechcraft 1900D was selected as the reference aircraft, since it was the most flown 19-seater commuter aircraft in Europe in 2019 [15]. The wing of the Beechcraft is dimensioned by the cruise flight, unlike many commuter or regional aircraft where the wing area is specified by the take-off case. Therefore, the influence of WTP and DP is also primarily considered in cruise flight. Under cruise conditions, the reference propeller with a diameter of \(d_p=2.79~m\) has a rotational speed of \(\Omega =1400~rpm\). Since DP and WTP require sufficient ground clearance, for reasons of CS 23 certification regulations, a high-wing redesign of the original Beechcraft 1900D was created in order to allow a fair comparison between the conventional and the novel configurations. In addition a straight wing leading edge in the outer part of the wing was chosen, to unify the inflow condition for different propeller positions. Since the exact geometry of the Beechcraft 1900D is not known, a simplified design of the wing was determined based on the literature data, see [15]. The wing airfoil used in the root area is a NACA 23018 and at the wing tip a NACA 23012. In the inner area, the airfoil was kept constant, and in the outer area, a linear transition from \(18\%\) to \(12\%\) thickness was applied. The inflow conditions in cruise flight are a Mach number of \(M = 0.4\) (\(v = 124.5 m/s\)) and a Reynolds number of \(Re = 1.15 \cdot 10e^7\) based on the wing root chord of \(c=2.45~m\).

The data of the modified conventional reference aircraft based on the Beechcraft 1900D are shown in Table 1.

Table 1 Technical data of the reference configuration
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2.1 Propeller distribution approaches

Based on the reference configuration, two different approaches to distribute the propellers were conducted, a full distribution of propellers over the entire span, which means that the entire wing is covered by propeller slipstreams, and a partial distribution with a limited coverage and a concentration of propellers on the outer part of the wing, see Fig. 2.

2.1.1 Full distribution

The number of propellers n for full DP can be estimated using a method by Hepperle [3]. The approach is a purely geometric estimation, which establishes a relationship between the number of propellers, their diameter \(d_p\) and the wingspan b of the reference aircraft. Hepperle’s approach, however, is extended here by subtracting the fuselage diameter \(d_f\) and a distance a between the propellers from the span that has to be covered. Furthermore, the position of the last propeller is set at the wing tip. The geometric constraint in this case is the given span:

$$\begin{aligned} b = d_f + 2\left( \frac{1}{2}~d_p + \left( \frac{n}{2}-1 \right) ~d_p +\frac{n}{2}~a \right) . \end{aligned}$$
(1)

The propeller diameter for a complete covering of the span with propellers results from this as a function of the number of propellers as follows:

$$\begin{aligned} d_p = \frac{b-d_f-n~a}{\left( n-1 \right) }. \end{aligned}$$
(2)
Fig. 2
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Distribution of propellers along the span

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As a consequence, the number of propellers thus determines the total propeller disk area

$$\begin{aligned} S_T = n~\frac{\pi ~d_p^2}{4}. \end{aligned}$$
(3)

The quotient of total to reference propeller disk area

$$\begin{aligned} \frac{S_T(n)}{S_{Ref}} \end{aligned}$$
(4)

allows for an aero-propulsive comparison between the different configurations. Following Hepperle, only for a certain number of propellers, the total propeller area is the same as for the reference configuration. The total reference disk area is achieved with 14 propellers, i.e., with \(n=7\) per half-span wing. With a reduction in the number of propellers to be distributed, the total propeller disk area is increased accordingly. For the parametric study full DP configurations in the range between 3 and 7 propellers are considered, which corresponds to a relative propeller disk area of \(\frac{S_T(n)}{S_{Ref}} \approx 1-3\). A small distance of \(a=15\%d_p\) between the individual propellers and to the fuselage is used.

2.1.2 Partial distribution

As an alternative approach, a partial distribution starting from the wing tip was formulated where the total propeller disk area is kept constant. Thereby the first propeller is a WTP and each additional one is positioned with a certain distance a after the other. The propeller diameter \(d_p\) thereby results directly from the number of propellers. The geometrical constraint here is the constant total propeller disk area \(S_T\), see Eq. 3. A constant total disk area allows for a pure comparison of the effect of the aerodynamic interaction on the required power without significantly chancing the propulsive efficiency for the different propeller designs. Rearranged to the diameter, the following results:

$$\begin{aligned} d_p = \root \of {\frac{4~S_{Ref}}{n~\pi }}. \end{aligned}$$
(5)

As a result, the number of propellers thereby dimensions the covered wing span:

$$\begin{aligned} b_{c} = d_f + \left( n-1 \right) ~d_p + \left( n-2 \right) ~a . \end{aligned}$$
(6)

The comparison parameter is the quotient of the span covered by propellers to the total reference wing span:

$$\begin{aligned} \frac{b_c(n)}{b_{Ref}}. \end{aligned}$$
(7)

For one propeller, the reference propeller is, therefore, placed at the wing tip and acts as WTP. With two propellers, each with half the propeller disk area and correspondingly reduced diameter, is positioned starting from the wing tip. For the study, the distance between the propellers is again set to \(a=15\%d_p\), and partial distributions between 1 and 5 propellers per half-span are considered, which corresponds to a coverage ratio of \(\frac{b_c(n)}{b_{Ref}} \approx 0.24-0.8\).

2.2 Propeller design

The propellers used in this study were designed with the in-house tool RAPID (Research Algorithm for Propeller Identification and Design) developed by the IFAS. The tool is based on an extended blade element momentum theory (BEM) as described by Adkins & Liebeck [16] and is enhanced by the extensions of Lück et al. [17]. The underlying theory will be summarized in the following. The relatively simple impulse theory consists of an axial and a rotational part. In the axial direction, the thrust of the propeller accelerates the flow. If an incompressible and inviscid flow is assumed, the thrust is derived from the mass flow through the circular propeller disk and the velocity difference between the undisturbed inflow and the propeller wake. Using Froude’s theorem and introducing the axial interference factor, the increase in axial velocity in the propeller disk can be calculated. Since the latter is directly related to thrust, the thrust can be described as a function of the axial interference factor. A similar approach for the swirl in the propeller wake leads to the rotational interference factor, which describes the induced rotational velocity in the propeller disk and behind it. Thus, finally, the torque of the propeller can be expressed as a function of the axial and rotational interference factors. The blade element theory (BET), which is used in addition to the momentum theory, applies the force calculation on wings to propeller blades. For this purpose, the propeller blade is divided into independent radial elements, to each of which an aerodynamic profile is assigned with the characteristics lift and drag coefficients at specific angles of attack (AoA). The generated force can be calculated via these coefficients, the chord length and the twist of the blade element. The velocity for this is determined from the superposition of flight velocity and the rotational velocity of the propeller. The integral propeller values for thrust and torque are finally obtained by integrating the sectional forces calculated in this way from the blade root to the blade tip. The combination of momentum theory and BET is then referred to as BEM. This requires that the calculated forces in the blade sections of the BET are equal to the forces of the momentum theory. For this purpose, the interference factors are taken into account in the inflow velocities of the sections. In addition, a loss factor that takes into account the finite number of blades and their size can be used. The Prandtl tip loss factor is often used for this purpose. In summary, the main task of the BEM is thus to calculate the forces on the propeller blade as well as the interference factors. If these quantities are known, the flow field is thus also described by the momentum theory.

Based on the data of the two distribution approaches, the reference propeller of the Beechcraft was scaled with a constant advance ratio to different diameters depending on the number of propellers and a propeller design with RAPID was performed by IFAS. The design data of the propellers for the full and partial DP approaches are (Tables 2, 3):

Table 2 Technical data of the full DP propellers
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Table 3 Technical data of the partial DP propellers
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2.3 Numerical setup

Parametric CFD studies were performed for the different full and partial DP configurations as well as the reference configuration, each with propellers positioned in front of a half-wing, neglecting the fuselage. The CFD simulations were performed by the IAG with TAU, the finite-volume flow solver developed by the German Aerospace Center (DLR), employing Reynolds-averaged Navier–Stokes (RANS) equations [18]. A second-order central scheme is used for spatial discretization and the turbulence is modeled with the Spalart–Allmaras turbulence model with rotation correction (SARC). All simulations are performed fully turbulent, neglecting a possible effect of the propeller slipstream on the laminar turbulent transition and thus an increased viscous drag due to reduced laminar lengths. Due to the low Mach number, preconditioning is used. The propellers are modeled using the TAU-internal steady-state Actuator Disk (ACD) method. Propeller polars are used as input data, which are generated by IFAS with RAPID. An algorithm developed at the IAG was used to achieve cruise flight conditions (lift-weight and thrust-drag equilibrium) in the simulations, see [15]. An iterative adjustment of the propeller rotational speed and the wing AoA is performed until constant cruise is reached.

2.3.1 Actuator disk

With the ACD the propeller blade forces are averaged over one propeller rotation and introduced stationary into the flow field on a circular boundary condition (BC), see Raichle et al. [19]. The TAU ACD generates non-uniform axial and tangential velocity profiles to achieve a physically correct representation of the propeller slipstream influence on the wing. A BET-based force calculation within the ACD takes into account the influences of the wing on the propeller. Thus, the numerical method is able to capture both the influence of the propeller slipstream on the wing and vice versa of the wing on the propeller forces. The local effective AoA and the local inflow velocity for the BET are directly determined from the CFD results at each point of the disk. The steady-state calculation enormously reduces the computation time compared to fully resolved propeller simulation. The ACD has, however, limitations as the slipstream is stationary and blade tip vortices are not taken into account. The time averaged influence of the interactions on the wing and propeller coefficients is, however, captured and comparable with advanced methods like Actuator Line (ACL) or fully resolved simulation [20,21,22].

2.3.2 Numerical grid

Fig. 3
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Half-span wing grid and ACD

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Fig. 4
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Isolated propeller grid

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For each propeller design, simulations are performed on two grids, one grid with a single isolated propeller and one grid with the corresponding number of propellers installed in front of the half-wing of the reference configuration. Figure 3 shows the half-span wing grid. A hybrid grid approach is used. Structured cells are utilized in the boundary layer (BL) of the wing. To ensure a low dissipation of the propeller slipstream, a quasi-Cartesian structured domain is used in the area, where the propellers are distributed. It extends from 0.7c in front of to 1c behind of the wing with a height of 1c. A refined area is used in the region of the wing tip vortex. The transition between the wing BL and the quasi-Cartesian domain is realized by unstructured cells. The structured domain is embedded in a refined unstructured domain with a radius of 1c. The latter domain is itself embedded in a coarser farfield area with a distance of up to 100c. A grid convergence studies with the same grid can be found in [15]. The airfoil of the final grid is discretized by 150 points in chordwise direction per upper and lower side and the BL by 45 structured cells and \(y^+ < 1\). The wing is discretized with 218 points in spanwise direction and the cell size of the quasi-Cartesian grid corresponds to \(2\%~c\). The vortex core is discretized with \(0.75\%~c\). In total, \(\approx 16-24\) mio cells are used, depending on the number of propellers. The isolated propeller grid, shown in Fig. 4, consists of the ACD BC and the same Cartesian propeller wake domain embedded in an unstructured spherical far field.

2.3.3 Cruise algorithm

In order to adjust cruise conditions (lift and weight \(L = W\) as well as thrust and drag \(T = D\) equilibrium) in the CFD simulation, an algorithm was used. Figure 5 shows the flowchart of the algorithm with the combination of two iteration loops. Inside of TAU, an AoA iteration exists to adjust a certain lift coefficient \(c_l\), which is used here for every simulation to obtain the lift and weight equilibrium. Furthermore, TAU is embedded in an additional iteration loop, that was programmed to reach the thrust and drag equilibrium. The cruise algorithm takes into account the influence of the AoA iteration together with the influence of the interactions on the wing drag in the thrust and drag equilibrium, while the other drag components (mainly fuselage) is given as constant from the aircraft design tool. A secant method is used to iterate the thrust, which is basically a zero point search for the function \(T - D = 0\). The rotational speed or the blade pitch can both be iterated to set the thrust. First, a TAU calculation is made with a starting value followed by reducing the current value by \(10\%\) and followed by another TAU calculation. This allows the secant method to start, since it requires two previous solutions. For the calibration of the isolated propellers the propeller blade pitch was adjusted until a given design thrust was reached to account for methodological differences in the local AoA estimation between the propeller polar generation method and the TAU ACD. For the DP and WTP parameter studies, the rotational speed was varied and a converged pre-calculation with constant AoA and constant rotational speed was conducted, before the start of the thrust iteration loops. The rotational speed was chosen to be varied instead of the pitch angle, because on the one hand, it was found that a reduced rotational speed increases the swirl recovery effect and on the other hand the propeller geometry was thus hold constant. In each loop, 1500 inner iterations were calculated at a constant rotational speed with the TAU-internal AoA iteration before the rotational speed was varied. In most cases only 2–3 loops are needed to reach a deviation from \(T = D\) equilibrium of \(< 0.2\%\).

Fig. 5
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Flowchart of the cruise algorithm

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3 Results and discussion

The parameter study on the number of propellers considers both the full and the partial DP approach, where the case with one propeller for the partial DP is at the same time the WTP case. The study, therefore, considers both DP and WTP under the same boundary conditions. All propellers are always simulated rotating against the wingtip vortex and the results shown are all from simulations at cruise conditions, unless otherwise stated.

3.1 Isolated propellers

Figure 6 shows a convergence study on the influence of the grid resolution of the ACD BC on the propeller coefficients. Both the tangential and radial number of points on the propeller disk were varied. With a deviation from the Richardson extrapolated value for \(c_t\) and \(c_p\) of less than \(1\%\), the selected resolution is considered to be sufficiently discretized. Figure 7 shows the propeller force in normal direction, i.e., the thrust, for the reference propeller and for exemplary partial and full DP propellers. For the partial DPs, the propeller load decreases with an increasing number of propellers, since the total disk areas is kept constant and the thrust is thus divided among more propellers. For the full DP propellers, the total disk area is approximately constant for a propeller number of \(n=7\). As the number of propellers decreases, the total disk area increases, so the isolated propeller load continues to decrease even though the thrust is distributed over fewer propellers. The propulsive efficiency of the isolated reference propeller is 0.91 and the efficiencies of the full and partial DP propellers is varying between \(0.90-0.92\)—thus almost constant.

Fig. 6
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Grid convergence study

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Fig. 7
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Radial distribution of propeller normal force

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3.2 Partial and full distributed propellers

The results of the study concerning the number of propellers are shown in Fig. 8. The propellers of the partial and full DP configurations are positioned with a distance in chordwise direction of 1.5 times the tip chord length before of the local leading edge. Thus the distance relative to the local chord length differs between the different propellers and the configurations. Figure 8a shows the effective aerodynamic efficiency, including the lift component due to the vertical propeller forces, as well as the required power, relative to the reference aircraft. The change in lift before the AoA iteration is shown in Fig. 8b, the change in drag in Fig. 8c. The results of the full DP are shown in red and the one of the partial DP in blue. For full DP, only small efficiency gains can be seen. The advantages due to the lift increase are transformed into drag savings after cruise conditions are reached due to an enabled AoA reduction. At constant AoA a lift increase occurs for all configurations due to the propeller blowing effect caused by the axial velocity in the slipstream. After cruise conditions are reached, this propeller provided lift increase leads to a reduction in AoA for the given lift-weight equilibrium, which corresponds to a lower wing drag as well. The full DP shows an increase in lift, Fig. 8b, due to the effect of the axial velocity in the propeller slipstream.

Fig. 8
figure 8

Parameter study of the number of propellers for partial and full DP

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However, the increase in viscous drag through the distribution along the whole span and the small reduction in induced drag due to the tangential velocity in the propeller slipstream, Fig. 8c, reduces the benefits. The lower the number of propellers, the higher the aerodynamic efficiency and the greater the reduction of the required power in cruise flight, Fig. 8a. For three propellers per half-span, the aerodynamic efficiency is maximal increased by about \(5\%\) and the required power is reduced by \(2.9\%\) relative to the reference configuration. For 6–7 propellers even a slight increase of the required power occurs. It must be noted that the full distribution approach according to Hepperle [3] is designed for regional aircraft with wings dimensioned by the take-off case, whereby higher propeller loads perform a higher lift increase and the wing area can then be reduced in the take-off case and thus be dimensioned for the cruise case. Since the wing area of the Beechcraft is already optimized for cruise flight and the propeller loads are significantly lower in cruise flight, the effects are not that pronounced here with a full DP. The results are in line with the cruise results by Keller [5], where also negligible performance changes were found for full DP with constant total disk area, without a reduction of the wing area due to take-off case improvements. In the pure cruise consideration here, a smaller number of propellers is advantageous for the full DP in multiple ways. First, a larger percentage of the total thrust is concentrated in the WTP, which achieves the best drag reduction. The positive effect of the higher thrust ratio in the WTP has already been shown in Keller [5]. Second, the viscous drag decreases, Fig. 8c, since the propeller load and thus the axial velocity in the propeller slipstream decreases due to the higher total disk area, Fig. 7. Third, the larger total disk area increases the propeller efficiency. As a consequence of the lower axial velocity, the lift increase is, however, also lower. The dependency of the lift from the propeller number is, however, rather weak, Fig. 8b. For a targeted reduction of wing area due to lift increase in the take-off case as described in Patterson [23] or Keller [5], an assessment could, however, be different, but for a pure cruise consideration, a smaller number of larger propellers is found advantageous. The benefits of a smaller number of propellers is continued with the partial DP. For a partial DP, larger increases in aerodynamic efficiency of up to \(14\%\) and larger reductions in required power of up to \(4.5\%\) are found, Fig. 8a. The aerodynamic efficiency is generally higher with the partial DP than with the full DP.

Fig. 9
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Full DP spanwise lift and drag distribution

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Fig. 10
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Partial DP spanwise lift and drag distribution

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The reduction in pressure drag, Fig. 8c, is more pronounced by the partial DP, due to the larger thrust ratio in the WTP and the concentration of the propellers at the wing tip area of the wing. The viscous drag is, however, lower for the full DP results than for the partial DP results, because the total disk area increases for \(n<7\) with full DP and thus the axial velocity is reduced. In comparison with a constant disk area (full DP \(n=7\) and all partial DPs), however, the viscous drag is lower for all partial DPs, since a smaller part of the span is covered. A direct comparison of partial and full DP for 3–5 propellers shows a higher aerodynamic efficiency increase with the partial DP, but the power requirement reduction is almost constant. The reason for this is the higher propeller efficiency for the full DP. Figure 8d shows the propeller efficiency of the individual propellers for different configurations. The propeller index 1 represents the propeller positioned at the wingtip, and each further index represents the following propeller. For the full DP with \(n=7\), the decrease of the efficiency through a positioning further outside is best shown. Due to the decrease in circulation towards the wingtip, the velocity reduction due to the wing’s present is lower at the respective propeller. The lower efficiency for the DPs positioned further outboard has already been shown in Keller [5]. The comparison of the full DPs shows the efficiency increase due to the increased total disk area, whereas for \(n=7\), only the innermost propeller has a higher efficiency than the reference propeller, for \(n=5\), there are already three, and for \(n=3\), just the WTP is lower. Due to the further outboard positioning of the partial DPs compared to the reference, the propeller efficiency is reduced in most cases. However, the decrease in propeller efficiency is overcompensated by the improvement in aerodynamic efficiency. It must be considered that the efficiency of the reference propeller is already high. With a less efficient reference propeller, the efficiency advantage with full DP by increasing the total disk areas could be more important. The parameter study shows that for wings dimensioned for cruise flight, more efficient configurations can be realized with partial DP than with full DP with the pure WTP configuration as the optimum of the propeller distribution along the wing.

Figures 9 and 10 show the spanwise lift and drag distributions for the full DP case with \(n=5\) and the partial case with \(n=3\). This comparison was chosen as an example because the propeller diameters are approximately the same for both cases. For comparison reason, the distribution of the isolated wing with the same total lift \(C_L\) is shown as well. Due to the smaller total disk area, the propeller load is larger for the partial DP, which is shown in the dimensionless thrust distribution in the front view of the configuration. The lift modification in the area of the outer three propellers is, therefore, more pronounced in the partial DP case than in the full DP case. For the same \(C_L\), the AoA for the partial DP configuration with \(\alpha =1.46\) is smaller than in the full DP case with \(\alpha =1.49\)—as a comparison the isolated wing reaches the same lift at \(\alpha =1.80\). The reduced AoA can as well be seen by the local lift and drag decrease in the root area of the wing for the partial DP. The concentration of the propellers on the outer area instead of a full DP is advantageous concerning drag in two ways. The pressure drag reduction is more pronounced due to the larger gradient of the circulation distribution in the outer wing area as well as through the higher thrust ratio in the WTP. In addition, the viscous drag is increased along a smaller range of the span.

The simulations were carried out fully turbulent and the influence of the propellers on the transition behavior is, therefore, neglected. Assuming a reduced laminar length in the area of a slipstream due to the propeller impact, a further advantage of the partial DP can be assumed under realistic flight conditions, since the root area of the wing can remain undisturbed and thus a lower transition related viscous drag can be expected for partial than for full distribution.

3.3 Wingtip propeller

A series of parameter studies was subsequently carried out for the pure WTP configuration in order to identify the influence of the following parameters:

  • Chordwise position x/c: The WTP was varied in the range 0.3x/c to 2.0x/c in front of the wing. The spanwise position was hold constant with y/(b/2)=1.0 and zero inclination.

  • Vertical position z/c: Moderate over as well as under the wing positions relative to the trailing edge height were considered up to a maximum of 0.2z/c each with zero inclination.

  • Spanwise position y/(b/2): The spanwise WTP position was varied between \(0.89< y/(b/2) < 1.03\) with a constant chordwise distance of \(x/c=1.0\) to the leading edge and zero inclination.

  • Inclination angle \(\beta\) of the WTP: The influence of an inclined propeller disk was considered in the range of \(-10^circ\) to \(+10^circ\). Negative values thereby describe a downward tilted propeller, positive an upward tilted one. The spanwise position was hold constant at \(y/(b/2)=1.0\) and the chordwise position at \(x/c=1.5\).

Fig. 11
figure 11

Parameter study of the WTP position and inclination angle

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Figure 11a shows the influence of chordwise position on the aerodynamic efficiency of the wing and the required power relative to the reference configuration. The value refers to the distance between the propeller and leading edge of the wingtip. The decisive parameter for the evaluation of the configuration is the required power, which remains largely constant. Only for the WTP positions closest to the wing, the required power is slightly increased. The power remains largely constant as the changes in aerodynamic efficiency and propeller efficiency balance each other out, see the generalized Munk’s stagger theorem [10]. With a further distance, the propeller stream is developed more before it reaches the wing, and the aerodynamic efficiency of the wing, therefore, increases. The propeller efficiency, however, decreases, since the velocity at a further distance is not reduced as much due to the presence of the wing compared with a closer distance. The independency of the required power from the chordwise propeller position also justifies the comparison of different DP configurations in Sect. 3.2 with different propeller distances from the wing. The relation of the global wing and propeller coefficients from the vertical propeller position is shown in Fig. 11b. The origin of z/c is at the height of the trailing edge. The power optimum is at the height of the trailing edge. While the lift increases for a higher position, the propeller efficiency decreases and vice versa the lift decreases for a lower position where the propeller efficiency is higher. Figure 11c shows the influence of the spanwise position. The spanwise power optimum is at \(y/(b/2)=0.95\) instead of at an exact wingtip position, which was already been found for other WTP configurations [22, 24]. The reason was found to be that a slightly inward WTP leads to an increased downward displacement of the wingtip vortex due to the WTP slipstream. The aerodynamic efficiency thereby increases due to the stronger downward vortex shift [22]. Both the aerodynamic efficiency and the propeller efficiency are higher here compared to positions further outboard. The inclination angle in Fig. 11d shows a strong dependency of both the aerodynamic efficiency and the required power. While the pure aerodynamic efficiency of the wing decreases with the propeller tilted upward, since the axial velocity in the stream creates a downwash, the effective aerodynamic efficiency increases, since a part of the thrust contributes to lift.

Fig. 12
figure 12

Ref spanwise lift and drag distribution

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Fig. 13
figure 13

WTP spanwise lift and drag distribution

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The propeller efficiency reaches its maximum at \(0^\circ\) and decreases rapidly with a downward tilting. In combination, the optimum of the required power of the configuration is achieved for a WTP inclined upward by \(3.75^\circ\). Analogous parameter studies with the partial DP configuration with two propellers per half-wing in [15] have led to the same results.

By combining the optimal parameter settings, an optimized WTP configuration has been derived with a WTP at \(y/(b/2)=0.95\) and an inclination angle of \(3.75^\circ\). The differences relative to the reference configuration, achieved with the optimized WTP configuration are shown in Table 4.

Table 4 Effect of the optimized WTP configuration
Full size table

Figures 12 and 13 show the spanwise lift and drag distribution of the reference and the optimized WTP configuration with the same total effective lift \(C_{L,eff}\). Again, the isolated wing is shown for comparison. The reference configuration achieves the lift at an AoA of \(\alpha =1.61^\circ\), while it can be reduced to \(\alpha =1.37^\circ\) by the optimized WTP configuration. The reference propeller has almost no influence on the global effective aerodynamic efficiency with \(\Delta C_L/C_D=0.1\%\) relative to the isolated wing. For the very inboard reference propeller position, the positive effect on the wing through the upwash side of the propeller slipstream are approximately balanced out by the negative effect of the downwash side. The gradient of the circulation distribution in the inner area is small. The pressure drag reduction is below \(\Delta C_{D,p}=-1\%\). The viscous drag increases in the area covered by the propeller slipstream. In total, a difference of \(\Delta C_{D,v}=2.9\%\) occurs.

In contrast, the optimized WTP shown in Fig. 13 increases the effective aerodynamic efficiency by \(\Delta C_L/C_D=17.0\%\) relative to the reference. The slightly inboard position optimally exploits the positive effect of the upwash in the propeller slipstream influencing the area covered by the wing and the wingtip vortex. The pressure drag is reduced by \(\Delta C_{D,p}=-21.5\%\) relative to the isolated wing. The propeller generates only minimal forces in the inner area, which can be seen in the front view of the configuration. For an even more inward position, the increasing downwash would start to increase the pressure drag in the outer region. The WTP position is also advantageous regarding viscous drag, since of all the configurations considered, the least part is covered by the propeller slipstream and increases only by \(\Delta C_{D,v}=0.5\%\) relative to the isolated wing. The comparison of the thrust distributions shows that the WTP has an asymmetric force distribution due to the fact that only one side is influenced by the wing. As can be seen in Fig. 8d, the propulsive efficiency of the WTP is slightly lower than the propeller efficiency of the reference configuration. However, due to the increase in aerodynamic efficiency the optimized WTP configuration is the most efficient configuration of all considered concepts.

4 Conclusion

A CFD-based parameter study of aerodynamically relevant concepts utilizing beneficial interactions between the propellers and the wing was carried out. For this purpose, different DP and WTP configurations were investigated and evaluated under the same boundary conditions. The investigation is based on a Beechcraft 1900D as a reference aircraft, modified as a high-wing version. Since the wing of the reference configuration is dimensioned by cruise flight conditions, the latter were used as the basis flow case for the investigations. For this purpose, an algorithm was used, iterating propeller rotational speed and lift AoA. For the distribution of the propellers along the span, two different approaches are compared. One with a full distribution along the span and one with a partial distribution with a concentration of the propellers on the outer part of the wing. It was found, that a pure WTP configuration enables a higher efficiency improvement than any DP configuration. For DP, it was found that at cruise flight, a partial distribution is more efficient than a full distribution. In the considered case, both approaches result in a more efficient configuration for a smaller number of propellers. It has to be stated, however, that the results are depending on the respective reference configuration. In the case of an aircraft with a wing dimensioned by take-off and an aimed improvement in efficiency by reducing the wing area, a different conclusion might be possible. A verification of the results for other reference configurations is, therefore, necessary. For the pure WTP configuration, the most important design parameters (positioning, inclination angle) were additionally varied in parameter studies. This resulted in the identification of the energetically optimal parameter combination for the WTP configuration, with which a reduction of the required power by \(\Delta P=5.6\%\) can be achieved. Since a pure WTP configuration is seen as problematic in terms of certification, either a configuration with a total of three propellers with two WTP and a propeller at the tail or a partial DP configuration with a total of four propellers is considered to be aerodynamically preferable. A comparison of these two approaches seems to be feasible, as a further step of the investigation.