## Abstract

Electrically driven ducted fans (e-fans), either underwing-mounted or located at the aft-fuselage, can potentially improve the system overall efficiency in hybrid-electric propulsion architectures by increasing their thrust share over the thrust generated by the main engines. However, the low design pressure ratio of such e-fans make them prone to operability issues at off-design conditions, i.e., takeoff, where nozzle pressure ratio is close or below the critical value. This paper investigates the operational limitations of such e-fans, proving the necessity of variable geometry. A component zooming approach is deployed by integrating a streamline curvature method within an aero-engine performance tool to investigate the e-fan installed performance and operability. The concepts of variable pitch fan (VPF) and variable area nozzle (VAN) are systematically explored to quantify any performance benefits, while the unavoidable added-weight challenges due to variable geometry are taken into account. Although e-fans with low design pressure ratio (PR) are more susceptible to operability issues compared to higher PR e-fans, the former show improved overall efficiency levels, mainly dominated by propulsive efficiency. It is found that variable geometry not only tackles operability but it can improve the off-design overall efficiency of e-fans even more. VPF mostly affects the component efficiency by reshaping the e-fan performance maps, while VAN has a greater impact on propulsive efficiency by moving the operating points.

## 1 Introduction

### 1.1 Background.

Current design trends indicate that electrically driven ducted fans (e-fans) play a critical role for the aero-propulsive integration benefits of hybrid-electric propulsion architectures, leading to increased effective bypass ratio and improved propulsive efficiency [1,2]. Parametric studies in terms of fan pressure ratio (FPR) and tip speed shown significant benefits on system performance and noise emissions [3,4]. Integrated sizing studies of e-fan and electric machine design in retrofitted aircraft applications revealed favorable design FPR levels between 1.27 and 1.35 [5–8].

Such low pressure ratio fans, though, are prone to operability issues at off-design takeoff conditions. The reason is twofold and can be explained by gas dynamics and the fan-nozzle matching, as described in detail by Kurzke and Halliwell [9]. It is true that nozzle pressure ratio determines the corrected massflow through a nozzle (choked versus unchoked). At takeoff conditions, nozzle pressure ratio is usually close or below the critical value (unchoked), decreasing the fan exit corrected massflow and therefore the takeoff operating line moves toward the surge line (reason 1). Moreover, a low FPR fan experiences much lower nozzle Mach numbers at takeoff compared to a higher FPR fan, reducing the exit corrected flow even more (reason 2). It should be reminded that lines of low exit corrected massflow lie above the lines of high exit corrected massflow in a conventional fan map. Thus, the takeoff point in a low FPR fan is much closer to the surge line than a cruise operating point (reason 1) or a high FPR fan takeoff point (reason 2).

Operability becomes even more challenging for fan designs that are located at the aft-fuselage, since they are exposed to highly distorted and nonuniform boundary layer flow surrounding the aircraft body. The so-called boundary layer ingesting (BLI) fans, however, can offer substantial reductions in fuel burn and pollutant emissions due to ingesting and re-energizing the low momentum fluid of the aircraft boundary layer [10]. Several published studies focus on BLI-tolerant fan designs using high-fidelity simulation [11–14], as a way to tackle the challenging internal flow. At a later phase, efficient and stable fan operation across the entire flight envelope should be ensured.

This is when variable geometry becomes necessary and can secure sufficient engine stability [15]. The concepts of variable pitch fan (VPF) and variable area nozzle (VAN) have prevailed as the most promising technologies. Their principles, though, are different. An change in rotor blade pitch to close the blade cascade moves the fan map characteristics to lower mass flow capacities, thereby increasing the fan surge margin without changing the fan operating point. On the other hand, opening the fan nozzle throat area allows more airflow passing through the fan, which moves the fan operating point to higher mass flow capacities and hence fan surge margin increases. However, variable geometry incur system complexity and weight penalties that could potentially negate the beneficial impact on installed system performance [16].

### 1.2 Variable Geometry Concepts.

Representative illustration of a variable pitch e-fan is shown in Fig. 1. The first report on exploring variable pitch solutions is originated in the early 1970s by NASA for the TF34 quiet engine concept [17]. In the early 1980s, NASA reported the first mechanical and aerodynamic analysis of the variable pitch concept as part of the parametric modulating flow fan (MODFAN) software [18,19]. The first two VPF technology demonstrators were presented by Hamilton Standard and joint NASA/GE forces, entitled Q-Fan [20,21], and Quiet Clean Short-haul Experimental Engine [22], respectively. Both programs provided high feasibility potential, since superior thrust-to-weight ratios and lower noise emissions at low altitudes were proven possible.

Halliwell and Justice [23] claimed that a clean sheet VPF engine could reveal the real potential of this concept, since VPF variant or derivative engines present aero-thermodynamic compromises for the engine core. The rotor blade pitch variation was implemented not only to prevent fan operability issues but also to reduce unnecessarily high surge margin and improve efficiency at high altitude conditions [24]. Typical pitch angle variations extended from $\u22125deg$ (opening the blade cascade) to $+10deg$ (closing the blade cascade) [20,23]. Analyses on how to generate VPF map characteristics are limited in the open literature [23,25] and installation effects due to added weight are usually neglected.

The concept of variable nozzle area has also been analyzed in the open literature, revealing significant benefits on engine performance and operation. Representative illustration of a variable area nozzle e-fan is shown in Fig. 2. Typical variations of nozzle throat area extended from −20% to +20% for both core and bypass nozzles in ultrahigh BPR turbofans [26,27]. Optimal overall efficiency and hence improved fuel consumption were accomplished by regulating nozzle geometry at cruise conditions [28–30]. Jet and fan broadband noise abatement were also possible by modulating fan nozzle throat during flyover or takeoff conditions in high BPR turbofan engines with low pressure ratio fan designs [31–34]. Similarly to the VPF case, the adverse installation effects due to nozzle variable geometry and the mechanism's added weight were neglected in the aforementioned studies.

A consistent comparison between VAN and VPF in terms of aircraft performance benefits was performed by Krishnan et al. [35]. It was concluded that turbofan engines with FPR lower than 1.4 benefit more from the VPF concept rather than VAN in terms of fuel burn, regardless of the overall pressure ratio. However, this study considered deterministic estimates of added weight for both concepts, as retrieved from Refs. [17] and [18]. The difference in fuel burn between VAN and VPF was reported in the order of 3%. The uncertainty of added weight levels, the neglected installation effects, and the standard scaled VPF maps from MODFAN [19] could impose questions on the confidence of the concluding outcomes. A follow-up work was presented by Aloyo et al. [36], revising the deployed methodology to include installation effects. This modification dictated that VPF could reduce block fuel more than VAN for any FPR design. It was still highlighted, though, that the confidence of this outcome was subject to the uncertainty of the assumed weight penalties.

Alexiou et al. [37] presented a design exploration study for ultrahigh BPR geared turbofan engines, using the concepts of VAN and VPF when necessary. A simplified approach to account the adverse installation effects of nacelle drag and drag-due-to-weight were employed. It was reported that the additional weight implications of both VAN and VPF mechanisms had the same order of magnitude, but the need of a more accurate weight estimation model to assess the engine installed performance was needed. Same conclusions on added weight were drawn by Mennicken et al. [38], who investigated potential operability improvements using the VPF and VAN, as well as oversizing the electric motor with respect to cruise power. Kavvalos et al. [25] presented a modeling approach and preliminary assessment of VPF and VAN concepts for low pressure ratio fans. An aero-thermodynamic analysis of rotor blade pitch variation was developed and integrated into a conceptual engine design framework. It was reported that a 30% surge margin (at constant mass flow) increase was achieved by closing the blade cascade by $8deg$, while 20% opening the nozzle area led to 25% surge margin improvement. Typical values for required compressor surge margin were reported between 10% and 20% [20,24,39], but there is not any universally accepted hard limit. Furthermore, it is often encountered that definitions of surge margin are ambiguous and not always suitable for any compressor component (multistage versus single-stage, high pressure ratio (PR) versus low PR).

### 1.3 Scope of This Work.

The objective of this study is to evaluate the impact of VPF and VAN on the installed off-design performance of low pressure ratio e-fans using a component zooming approach, where a performance tool is closely coupled with a two-dimensional (2D) streamline curvature fan model. This study has explored e-fan designs that are located at the aft-fuselage of a 19-passenger commuter aircraft, which incorporates a series/parallel partial hybrid architecture with the main engines (turboprop) mounted under-wing. The location of the e-fans at the aft-fuselage deemed necessary to consider the incoming momentum deficit due boundary layer ingestion. It should be highlighted, though, that the described component zooming approach and introduced metrics are considered configuration-agnostic.

Thus, a generic e-fan configuration has been established within the aero-engine performance tool EVA [40]. The DLR's in-house streamline curvature (SLC) code [41,42] has been deployed to realize three ducted e-fan designs with different pressure ratio, while keeping work coefficient fixed and meridional Mach number constant throughout the stage. Accurate off-design performance maps are generated with emphasis on a consistent surge line derivation. Feeding the tailored-for-each-application maps back to the performance tool improves the accuracy of off-design performance and operability prediction. The assessment is extended to account for adverse installation effects and individual efficiency contributions that are applicable to novel propulsion system architectures. The applicability of surge margin definitions on low pressure ratio fans has been extensively discussed using analytical equations.

The operational challenges of e-fans at low-speed and low-altitude off-design conditions constitute the main case study. Design fan pressure ratio and electric power input at end-of-runway takeoff are varied, and the necessity of variable geometry for an efficient and safe operation is revealed. To this extend, the effects of VPF and VAN on the installed performance and operability of e-fans have been systematically investigated. Rotor pitch settings of 2 deg and 4 deg (closing the blade cascade) and nozzle area opening of 5% and 10% have been examined, drawing clear conclusions on which design FPR levels benefit more from the variable geometry concepts. The uncertainty of the added weight for both mechanisms has been also considered, shedding light on whether weight and complexity could outweigh the benefits.

## 2 E-Fan Design

A systematic fan design rationale has been followed to allow for fair and consistent outcomes of the upcoming e-fan performance studies. Three e-fan cases are considered with pressure ratio of 1.2, 1.25, and 1.3 at the aerodynamic design point (DP). The design point is top of climb with flight Mach number of 0.32 at an altitude of 10,000 ft, as emerging from the top-level aircraft requirements of a 19PAX commuter aircraft. Electric power input at DP is fixed at the same value for all e-fan cases. The design mass flow values, as depicted in Table 1, emerged from the cycle performance study and component matching for each e-fan case.

Parameter | Units | FPR = 1.2 | FPR = 1.25 | FPR = 1.3 |
---|---|---|---|---|

W | kg/s | 30.5 | 24.8 | 20.9 |

η_{is} | % | 93.7 | 93.6 | 93.4 |

$\Phi $ | 0.760 | 0.695 | 0.650 | |

D_{tip} | m | 0.870 | 0.773 | 0.700 |

$Uc,tip$ | m/s | 206.3 | 226.9 | 249.8 |

Solidity (R,S) | 1.14, 1.18 | 1.32, 1.32 | 1.5, 1.47 | |

$\delta /hR$ | 1.51 | 1.71 | 1.87 | |

Weight | kg | 210 | 160 | 140 |

Parameter | Units | FPR = 1.2 | FPR = 1.25 | FPR = 1.3 |
---|---|---|---|---|

W | kg/s | 30.5 | 24.8 | 20.9 |

η_{is} | % | 93.7 | 93.6 | 93.4 |

$\Phi $ | 0.760 | 0.695 | 0.650 | |

D_{tip} | m | 0.870 | 0.773 | 0.700 |

$Uc,tip$ | m/s | 206.3 | 226.9 | 249.8 |

Solidity (R,S) | 1.14, 1.18 | 1.32, 1.32 | 1.5, 1.47 | |

$\delta /hR$ | 1.51 | 1.71 | 1.87 | |

Weight | kg | 210 | 160 | 140 |

The e-fan designs are carried out using the DLR's in-house SLC method and the main assumptions are summarized below:

Rotor and stator local diffusion factor are kept below 0.5 and de Haller above 0.72, as retrieved from textbook guidelines [43,44] and in-house design expertise [38,45].

A stator exit condition without remaining swirl is chosen, as remaining swirl directly translates into thrust losses.

The hub-to-tip ratio is kept constant to 0.72 for all e-fan designs, allowing for tip diameter to change in order to meet the required design mass flow and pressure ratio.

A constant stage work coefficient ($\Psi =\Delta ht/Utip2$) of 0.5 is selected, resulting in individual tip speeds for each e-fan, since work input was predefined by the user and the cycle performance.

An axisymmetric incoming boundary layer profile in terms of total pressure ratio is applied, based on the turbulent flat plate theory and the $1/7th$ power law [46]. Top level parameters (flight Mach and altitude) as well as representative fuselage dimensions are taken into account to have a first-order estimate of the boundary layer thickness and incoming momentum deficit for all e-fan designs. The former is nondimensionalized with rotor blade height ($\delta /hR$) and is presented in Table 1. Discussion on the aeropropulsive benefits and challenges of boundary layer ingestion on e-fan design lies outside the scope of this work, but is well described by Schnell et al. [7] and Mennicken et al. [38], following similar design rationale.

Blade count for rotor and stator are kept constant to 16 and 28, respectively, for all e-fan cases.

Constant averaged meridional Mach number levels of 0.41 are targeted to ensure design coherence among the different pressure ratio e-fans. Averaging is necessary at fan entry due to the incoming boundary layer profile and the subsequent rising Mach number levels over the span. The resulting flowpath with meridional Mach number isolines are depicted in Fig. 3 for the representative case of FPR = 1.2.

The last two assumptions drive the design strategy for this study. For all three designs, there is a fixed power input but a change in design FPR and mass flow. Fixed power input translates to certain thrust target when considering the propulsive efficiency of each fan design. Thus, fan dimensions do change and so does the pitch (=chord/solidity), as shown in Table 1. This is the main reason for the variation in solidity. In conjunction with the lower FPR, work input and thus flow turning decreases (Fig. 4(d)). Making the design decision of similar meridional Mach distribution for all FPR cases leads to different levels of flow deceleration, and hence De-Haller as shown in Figs. 4(b) and 4(e). The levels of diffusion factor are quite close for all three FPR cases, especially in the highly loaded stator hub region (Fig. 4(f)). This design philosophy and the subsequent effects on aerodynamic loading were considered sufficient to capture the first-order effects on operability limitations during an off-design performance assessment, which was the main focus of this study.

The radial distribution of major rotor and stator design parameters is presented in Fig. 4 for all examined e-fan cases. The low momentum region due to the incoming boundary layer justifies the radial increase of meridional Mach number at rotor inlet, as shown in Fig. 3. The authors followed the practice of “strong hub profile” in order to re-energize the boundary layer [38,47]. Said that, total pressure ratio increases at rotor hub, as observed in Fig. 4(a), resulting in a homogeneous total pressure profile at stage exit. The resulting smaller velocity gradients at stage exit have a beneficial effect on propulsive efficiency and downstream flow [48]. The resulting flow coefficient levels ($\Phi =Vx/U$) and rotor/stator solidity values are shown in Table 1. Comparable performance was achieved for all e-fan cases, with levels of isentropic efficiency very close to each other. The estimated weight for the three e-fan designs are presented in Table 1 including all e-fan components (blades, disks, casing, shaft, nacelle, and nozzle) without considering the electrical components (motor and power electronics).

An increase in De-Haller at rotor hub is observed in Fig. 4(b). The low circumferential velocity (due to hub radius) leads to a necessary increase in flow turning at rotor hub and a subsequent reduction in the exit relative swirl velocity at rotor hub to meet the required pressure ratio. However, the flow is accelerated at rotor hub exit leading to a substantial rise in axial velocity. The latter effect dominates the decrease in exit relative swirl velocity and explains the increased De-Haller at rotor hub. The accelerated flow has an adverse effect to the stator flow at hub as well. High inflow velocities at stator hub lead to high required flow turning in order to achieve a stator axial outflow. This translates to high diffusion factor and low De-Haller at stator hub, as shown in Figs. 4(f) and 4(e).

Once the e-fan designs were realized, off-design performance maps were generated using the SLC algorithm. It should be highlighted that extensive validation studies of the DLR's in-house SLC algorithm against experimental data and high-fidelity three-dimensional CFD simulation have been carried out the past decade for single-stage fans and multistage core compressors [49]. Consistency in the definition of surge line for all FPR cases was critical in order to ensure consistent operability-based findings. Thus, the strategy for defining the lower mass flow points of each speedline (assumed as surge line in this work) is set by a minimum De-Haller threshold of 0.65. Apart from consistency, this criterion is chosen to ensure reasonable levels of diffusion throughout the fan operating map, which is in-line with the limitation of the 2D SLC method to accurately model secondary flow phenomena based on real flow physics [50]. Furthermore, an inherent convergence criterion of the SLC method to meridional Mach numbers lower than 0.9 reveals the limitation of accurately modeling the choking region. The resulting e-fan off-design performance maps for FPR = 1.2, 1.25, and 1.3 are demonstrated in Figs. 5(a)–5(c), respectively.

## 3 Performance and Operability Metrics

This sections presents all metrics and figures of merit that will be used in the interpretation of performance results. It should be highlighted that all performance and operability metrics assume averaged states due to the conceptual nature of the study.

### 3.1 Transfer Efficiency.

### 3.2 Propulsive Efficiency.

It is defined as the ratio of useful uninstalled propulsive power over the net jet kinetic power. It should be noted that, even if the e-fan is operating under the presence of boundary layer ingestion, the propulsive power term must be multiplied by the clean freestream velocity, *V*_{0}. However, the e-fan net jet kinetic term should include the BLI velocity in order to calculate the incoming jet kinetic energy.

### 3.3 Installation Efficiency.

### 3.4 Overall Efficiency.

It can also be proved that the overall efficiency is the product of e-fan transfer, propulsive, and installation efficiencies.

### 3.5 Surge Margin.

## 4 Operational Limitations of Conventional E-Fans at Take-Off

Having generated the off-design maps of all e-fan cases in a consistent manner, the e-fan installed performance at off-design conditions is investigated. The term “conventional” refers to e-fans without any variable geometry. End of runway takeoff is analyzed in this study, since it is an important operating condition in terms of operability. The electric power input at end-of-runway takeoff was varied and expressed as a percentage value relative to the fixed power input at DP (which value is identical for all three e-fan designs). The parametric range for takeoff relative power extends from 50% to 150%, and is used in the abscissa of the following figures. For a fixed electric power at DP, the following research question rises:

How do take-off power and variable geometry affect the installed performance and operability of low pressure ratio e-fans?

The discussion on the validity of various surge margin definitions for low pressure ratio fans, which was presented in Sec. 3, is complemented by Fig. 7. Each surge margin definition is quantified for the investigated takeoff study. Figures 7(a) and 7(d) confirm the initial statement that surge margin definitions based on Eqs. (5) and (8) can be misleading for low pressure ratio fans and will be omitted from this study. The definition based on Eq. (6) presents higher surge margin levels compared to Eq. (7) due to the different denominators. The former considers the choking point in the denominator, while the latter sets a unity instead. As explained previously, accurate and consistent modeling of the choking region cannot be guaranteed with an SLC method, and therefore the definition of surge margin in Eq. (7), $SMcW,unity$, will be used as the default operability metric to retain consistency throughout the study.

The e-fan with FPR = 1.3 presents higher surge margin levels compared to lower FPR cases due to higher tip speeds and thus steeper fan characteristics, as depicted in Fig. 5. Nevertheless, the topology of points for different takeoff power input levels are demonstrated in Fig. 5 with black square markers, revealing a short distance to the surge line, although Fig. 7 shows a surge margin range of 10–20% which could be considered acceptable according to the literature. This contradiction is exactly the reason why it was deemed necessary to include and compare all different surge margins definitions at first hand. There are preliminary design studies in the literature where surge margin is often quoted without being clearly defined leading to inconsistent comparisons among different studies. In this work, it is observed that variable geometry is necessary and the threshold of 10–20% is not valid for the chosen surge margin definition.

The effects of design FPR and relative takeoff power input on individual e-fan efficiencies are presented in Fig. 8. It should be reminded that the off-design performance is assessed in this section, thus all efficiencies refer to takeoff conditions, and not the design point. As depicted in Fig. 8(a), transfer efficiency is slightly increasing with takeoff power due to the increasing mass flow and increasing ideal jet velocity levels. Although FPR = 1.2 allows for higher mass flow capacities, its lower ideal jet velocity levels (being a squared term in Eq. (1)) lead to a constant offset of approx. 5% compared to the rest two cases. Additionally, the points with different takeoff power for the case of FPR = 1.2 lie on a much lower component efficiency isoline compared to the 1.25 and 1.3, as observed from the maps in Fig. 5(a). The propulsive efficiency presented in Fig. 8(b) is relatively low compared to common trends due to low flight Mach of 0.32 which was a fixed top level parameter for this study. As expected, the higher mass flow capacities of FPR = 1.2 presented higher propulsive efficiency. Propulsive efficiency decreases for higher takeoff power input due to the increasing squared term of ideal jet velocity (since mass flow can be omitted from both numerator and denominator in Eq. (2)).

Figure 8(c) demonstrates that installation efficiency for higher design FPR levels outcompetes the lower ones. Fan diameter is the hidden driver, leading to milder installation effects for higher design FPR (and thus smaller fan diameter as included in Table 1). The increasing trend of installation efficiency with higher takeoff power lies on the fact that individual drag components are nearly constant for varying take off-power, while net thrust is rising and hence decreasing drag contributions (ratio of $FD/FN$ for nacelle form, nacelle skin friction, intake spillage, nozzle, and weight drag components). Since all drag contributions are subtracted from unity, as revealed in Eq. (3), installation efficiency is rising for higher takeoff power levels.

Finally, the product of all the aforementioned individual efficiencies provides the trend of e-fan overall efficiency, as shown in Fig. 8(d). It can be deduced that propulsive efficiency dominates the behavior in overall efficiency in terms of both design FPR and takeoff power levels. The metric of overall efficiency practically means how much of the provided electric power to the e-fan translates to installed propulsive power at takeoff conditions, revealing an order of 30–38%. If freestream velocity is omitted from the numerator of overall efficiency (Eq. (4)), a similarly useful figure of merit emerges; namely, the ratio of installed net thrust to power input. This figure of merit shows how much net thrust can be produced by a given power input (in (N/kW) units), considering all installation effects.

It can be argued that overall efficiency or the aforementioned ratio is less common objectives to maximize during takeoff compared to surge margin. However, these metrics play a major role in hybrid-electric propulsion systems where the presence of e-fans can enable engine downsizing. The thrust split between gas turbine engines and e-fans in a hybrid-electric propulsion system will determine the sizing of the engines. Maximizing e-fan net thrust for a given power input, or minimizing e-fan power input for the required net thrust, can unlock the potential of hybrid-electric and distributed propulsion systems. This is why overall efficiency and surge margin are the final metric to assess installed performance and operability for both conventional and variable-geometry ducted e-fans.

Therefore, the following messages should be communicated from this section: (i) Although e-fans with low design PR are more prone to operability issues compared to higher PR e-fans, they present higher overall efficiency which is mainly dominated by the propulsive efficiency; (ii) variable geometry seems essential for a safe takeoff operation regardless of the chosen FPR level and the provided end-of-runway takeoff power input, and (iii) choosing a high takeoff power input could enable substantial engine downsizing opportunities but it will cost in terms of e-fan overall efficiency by a maximum of 8%.

## 5 Operational Benefits and Challenges of Variable Geometry

In light of the low achievable surge margin levels for the conventional cases, the concepts of VPF and VAN are systematically investigated in this paper. Potential improvements or penalties on operability and off-design installed performance are quantified for the different e-fan designs. Several surge margin thresholds are listed in the open literature [25,43,52] for fan and compressor components, ranging from 10% to 20%. Considering the fact that the minimum threshold is often component- and application-specific, the authors decided to interpret the variable geometry results without commenting on a minimum acceptable limit. Therefore, all results are expressed in differences, $\Delta SM(=SMVG\u2212SMconv)$ in (%) and $\Delta \eta (=\eta VG\u2212\eta conv)$ in (%), between the variable geometry (VG) fan and the conventional (conv) fan. For further clarification, a positive $\Delta \eta $ or $\Delta SM$ value implies that variable geometry imposes an efficiency benefit or surge margin improvement, respectively, while a negative $\Delta \eta $ value introduces a penalty. The surge margin definition used in the following plots is based on Eq. (7) and denoted as SM.

### 5.1 Approach for Variable Pitch Fan Map Derivation.

A change in rotor pitch angle relocates the fan map characteristics to different mass flow, efficiency and total pressure ratio levels. Halliwell and Justice [23] proposed simple deltas in mass flow and efficiency with pitch angle, while suggesting that pressure ratio can be considered insensitive to rotor pitch variation for performance and conceptual design studies. An analytical approach to derive rapid VPF maps was developed by Kavvalos et al. [25], where they used a velocity triangle analysis to provide updated mass flow and efficiency deltas. Their employed approach assumed (i) operation along the backbone curve (which is the operating line connecting the maximum efficiency or minimum incidence points of all speedlines) and (ii) unaffected total pressure ratio with pitch variation. The reality is that a change in rotor pitch at off-design does affect total pressure ratio due to additional losses, especially when you move away from the backbone curve.

Thus, this part will analyze the aforementioned assumptions using the SLC method and comment on their validity for performance and conceptual design studies. A comparison between the VPF maps derived from the SLC algorithm (noted as “SLC”) and the VPF maps derived from the analytical approach proposed by Kavvalos et al. [25] (noted as “analytical”) is carried out. The effect of additional VPF-driven pressure losses on surge margin is investigated. It should be noted that the comparison focuses only on the isolated rotor and stator pressure losses (and their origin), by considering that both map derivation approaches apply the same deltas in mass flow for a given rotor pitch setting. Further information on how these deltas are derived can be found in Ref. [25] and will not be detailed here.

Figure 9 compares the mass-averaged velocity triangles at rotor inlet and rotor outlet between a conventional (solid black line) and a variable pitch fan (dashed gray line) with closed blade cascade (VPF > 0). The VPF analysis is interpreted for constant rotational speed, as shown in the respective figure. Closing the blade cascade (or increasing the pitch angle) reduces the mass flow capacity through the fan and therefore axial velocity decreases. For constant blade speed, VPF rotor inflow and outflow angles increase, while retaining constant swirl velocities. So, velocity triangles demonstrate clearly the effect of VPF on fluid kinematics, implemented on both SLC and analytical approaches. However, additional pressure losses due to rotor restaggering are not considered in the analytical approach. On the other hand, they are considered with the SLC method. Specifically, 2D cascade losses, secondary losses, and tip clearance losses are computed based on incidence and deviation models that are validated against three-dimensional CFD simulation for state-of-the-art compressors [41].

Total pressure loss coefficient, *ω*, is used for this analysis and is defined as the total pressure ratio difference normalized by the inlet dynamic head. Figure 10 illustrates the difference in loss coefficient, $\Delta \omega (=\omega SLC\u2212\omega an)$ in (-), between the SLC and the analytical approach. A positive $\Delta \omega $ value means that additional VPF-driven total pressure losses are imposed compared to the analytical approach. The analysis is carried out for two FPR cases (1.2 and 1.3) and four points in the map; the backbone (maximum efficiency) and surge (minimum mass flow) point at design speed (100%) and low speed (50%).

It is shown that total pressure losses along the backbone points are very low when changing the rotor pitch, which verifies the suggested approach by Kavvalos et al. [25]. The effect of blade row, design FPR, and rotational speed seem to be negligible for the backbone-based analysis. However, substantially increased losses are observed when performing the analysis for the point of minimum mass flow (named as surge point). The change in rotor pitch translates into rotor and stator incidence, superimposed on the already nonoptimal incidence of the surge point, leading to a substantial increment in 2D cascade (or profile) losses. It is observed that the adverse effects due to high incidence are more dominant in the stator rather than the rotor. The 2D cascade losses seem to be the main source of losses when changing the rotor pitch angle, presenting a >80% share compared to the rest two (secondary and clearance) for all tested cases. The effect of rotational speed does not play an important role, but some discrepancies are depicted for different design FPR levels.

The fan performance maps of VPF = 2 deg were derived with both approaches and the effect on surge margin improvement $\Delta SM$ is quantified in Fig. 11 for the same takeoff study. As expected, the total pressure losses move the SLC-based map to lower pressure ratio levels, and thus surge margin is reduced compared to the analytical one. However, the discrepancy in surge margin improvement between the SLC and the analytical approaches reaches a maximum value of 5% for any operating point. That value is considered acceptable and the assumption of unaffected total pressure ratio with pitch variation is reasonable for performance and conceptual design studies. Therefore, the following VPF results will be based on fan maps that are derived with the analytical approach.

### 5.2 Comparison of Variable Geometry Concepts.

The off-design study with variable end-of-runway takeoff power levels and different design FPR is now carried out for the variable-geometry e-fans, investigating potential performance and operability benefits. Surge margin improvement and individual efficiency deltas compared to the conventional fan cases are used to interpret the results. Four variable geometry instances are analyzed; rotor pitch of 2 deg and 4 deg, and nozzle area opening of 5% and 10%, being representative for such e-fan configurations [25,38]. The operability improvement with variable geometry can be detected in Fig. 12 by observing the topology of points with different takeoff power (black square markers) for the FPR = 1.2 case. It is evident that all points are now located in a relatively long way off the surge line for both VPF = 2 deg (Fig. 12(a)) and VAN = 10% (Fig. 12(b)), when compared against the conventional fan in Fig. 5(a). It is reminded that VPF > 0 relocates the fan map to lower mass flow values while VAN > 0 moves the operating point to higher mass flow values, and both lead to an increment in surge margin.

A correlation of how much rotor restaggering can achieve similar surge margin improvement as a certain opening of the fan nozzle area can be extracted by Fig. 13. First, it is shown that closing the rotor cascade by 2 deg improves surge margin by 10–15% depending on the chosen design FPR, while closing by another 2 deg reaches a benefit of 20–25%. On the other hand, opening the area nozzle by 5% results in a 7–8% improvement in surge margin, while opening by another 5% leads to an increment of 13–15%. It can be observed that surge margin improvement due to VPF is more sensitive to the chosen design FPR level (lines are spread more) compared to the VAN concept (lines are close to each other). Having said that, the low FPR designs (e.g., 1.2) seem to benefit more from a given variable geometry setting compared to higher FPR designs (e.g., 1.3), without implying that the absolute surge margin values of FPR = 1.2 are higher than the 1.3 case.

For quantifying the impact of variable geometry on individual efficiencies, the instances of VPF = 2 deg and VAN = 10% are selected, since they provide similar surge margin benefits (10–15%). It is evident from the updated maps in Fig. 12 that all operating points with different end-of-runway takeoff power levels lie in a relatively high-efficiency isoline now. The detailed effect of variable geometry on fan isentropic efficiency is illustrated in Fig. 16 in Appendix section. For the illustrated case of FPR = 1.2, isentropic efficiency rises for both VAN = 10% and VPF = 2 deg which directly translates to pronounced benefits in transfer efficiency when variable geometry is deployed, as shown in Fig. 14(a). Slightly higher benefits in isentropic efficiency are observed for the VAN case when compared to the corresponding change in VPF.

Moreover, propulsive efficiency (Fig. 14(b)) can be considered nearly unaffected for the VPF case, since it is matter of ideal jet velocity. It is the opening of area nozzle that increases mass flow and reduces ideal jet velocity compared to the conventional case, leading to a small benefit in propulsive efficiency of approx. 1%. As expected, installation efficiency (Fig. 14(c)) deteriorates for all variable geometry cases due to the added weight and hence increased weight drag (considering a 10% weight penalty for both variable geometry concepts). Furthermore, the effect of variable geometry on overall efficiency is depicted in Fig. 14(d) and the pronounced benefit of e-fan with design FPR = 1.2 holds true due to considerably higher component and transfer efficiencies compared to the conventional cases. It should be noted that VAN seems to perform slightly better than VPF over the entire range of takeoff power levels due to the higher propulsive efficiency. However, the efficiency deltas remain constant for different end-of-runway takeoff power levels (horizontal lines), retaining the same trends and outcomes as for the conventional cases.

Therefore, the following can be concluded from this section: (i) low FPR designs benefit more from variable geometry compared to higher FPR designs in terms of both surge margin and efficiency, (ii) closing the blade cascade by 2 deg demonstrates similar surge margin improvement with opening the nozzle area by 10% (revealing a correlation ratio of 5% to 1 deg), and (iii) VPF mostly affects the component efficiency by reshaping the e-fan performance maps, while VAN has a greater impact on propulsive efficiency by moving the operating points.

### 5.3 Effect of Uncertainty on Variable Geometry Weight.

The question of whether added weight and complexity of variable geometry outweigh its performance benefits is not clearly answered in the open literature. From a manufacturing perspective, both mechanisms are not very mature leading to substantial uncertainty on the levels of added weight that should be considered on performance and conceptual design studies. In Sec. 5.2, a penalty of 10% on fan system weight on top of the values included in Table 1 was considered for both VAN and VPF concepts (noted as *nominal* according to Ref. [36]). However, two more cases of weight penalties are explored to demonstrate if the above performance outcomes still hold true; (i) a best-case scenario of 5% and (ii) a worst-case scenario of 20% added weight.

The variable geometry added weight affects installation efficiency (solely driven by weight drag) which propagates to overall e-fan efficiency. Propulsive and transfer efficiency as well as surge margin remain insensitive to added weight variations. Figure 15 quantifies the uncertainty of added weight for VPF = 2 deg and VAN = 10%, in the form of error bars for the best- and worst-case scenario (notation explained in Fig. 15(c)). Moving from the best-case scenario (5% added weight) to the worst-case (20% added weight) can cost up to 1.2% in installation efficiency and 0.5% in overall efficiency for low takeoff power levels and low FPR designs. The cost for higher takeoff power levels and higher FPR designs is reduced to 0.5% in installation efficiency and 0.2% in overall efficiency. So, it can be concluded that the outcomes mentioned in this study remain quite robust even in the presence of added weight uncertainty for the variable geometry mechanisms.

## 6 Conclusions

This paper has conducted a systematic investigation for the impact of VPF and VAN on the installed performance and operability of low pressure ratio ducted e-fans. A component zooming approach has been deployed by integrating a SLC algorithm within an aero-engine performance tool. Three e-fan designs with different pressure ratio (FPR) have been realized, retaining design consistency in major aerodynamic parameters. Tailored off-design maps have been derived and then fed back to the performance tool. Adverse installation effects, surge margin definitions, and individual efficiencies (transfer, propulsive, installation, and overall) have been discussed in detail with analytical equations. VPF maps have been derived using both the SLC method and an analytical approach, proving that the usually neglected VPF-driven total pressure losses is reasonable for performance studies.

The major findings of this study are outlined below:

Low pressure ratio fans are prone to operability issues at off-design take-off conditions, revealing the necessity for variable geometry.

An e-fan with design FPR of 1.2 benefits more from variable geometry compared to higher FPR designs (1.25 and 1.3) in terms of both surge margin and overall efficiency. It is propulsive efficiency that mainly dominates the behavior of overall efficiency.

Comparing VPF against VAN, it is shown that closing the blade cascade by 2 deg leads to similar surge margin improvement with opening the nozzle area by 10%, revealing a correlation ratio of 5% to 1 deg.

The effect of uncertainty on variable-geometry added weight shown that the performance trends and findings of this study hold true for reasonable assumptions of weight penalties, since variations in installation and overall efficiency are negligible.

## Acknowledgment

The authors would like to acknowledge Dr. Markus Schnös for the continuing support with the streamline curvature algorithm and the valuable technical discussions.

This work has been partially financed by the research project E-THRUST and the THEMIS.

## Funding Data

The research project E-THRUST, funded by the Swedish Energy Agency (pr. no. 52415-1; Funder ID: 10.13039/501100004527).

The research project THEMIS, funded by the Knowledge Foundation (pr. no. 20200260; Funder ID: 10.13039/501100004527).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*c*=corrected

- chok =
choking region

*cN*=constant rotational speed

*cW*=constant mass flow

*i*=installation

- id =
ideal conditions

- in =
inlet conditions

- inst =
installed

- int =
intake component

- is =
isentropic

- kin =
kinetic

- nac =
nacelle component

- noz =
nozzle component

- out =
outlet conditions

- ov =
overall

- prop =
propulsive

*t*=total conditions

- tran =
transfer

- uninst =
uninstalled

*x*=axial direction

- 0 =
freestream conditions

### Appendix

The effect of variable pitch fan and variable area nozzle on the e-fan isentropic efficiency for different design FPR levels and takeoff power input levels is illustrated in Fig. 16.