2.1.1 Figures of merit

To evaluate the differences in BL properties, the BL area ${\rm{\Delta }}$ given in Equation (1a), its mass displacement area ${{\rm{\Delta }}^{\rm{*}}}$ , given in Equation (1b), and its kinetic energy area ${{\rm{\Theta }}^{\rm{*}}}$ in Equation (1c) are used. In comparison with the conventional 2D thickness definitions, these areas are applicable to 3D axi-symmetric scenarios whereby a local width parameter $b$ is introduced [Reference Drela21] and illustrated in Fig. 4. This parameter relates the 2D and 3D definitions by considering the circumferential area at any radial position.

(1a) \begin{align}{\rm{\Delta }} &= \mathop \smallint \nolimits_0^\delta \,b\left( n \right){\rm{d}}n\end{align}

(1b) \begin{align}{{\rm{\Delta }}^*} &= \mathop \smallint \nolimits_0^\delta \left( {1 - \frac{{\rho V}}{{{\rho _e}{V_e}}}} \right)b\left( n \right){\rm{d}}n\end{align}

(1c) \begin{align}{{\rm{\Theta }}^{\rm{*}}} &= \mathop \smallint \nolimits_0^\delta \left( {1 - \frac{{{V^2}}}{{V_e^2}}} \right)\frac{{\rho V}}{{{\rho _e}{V_e}}}b\left( n \right){\rm{d}}n\end{align}

where $\delta $ quantifies the BL thickness in a 2D cut plane, $\rho $ the fluid density and $V$ its velocity and where $e$ characterises the BL edge properties.

Figure 4. 2D and 3D axi-symmetric boundary layer parameters at cut planes around the fuselage tail, inspired from Ref. [Reference Lamprakis, Sanders and Laskaridis16].

Along with the development of the BL properties, common mechanical energy-rate terms present in the energy- and exergy-based analyses [Reference Sanders and Laskaridis4, Reference Arntz, Atinault and Merlen9, Reference Drela10] are calculated on survey planes along the fuselage. In doing so, the power balance is given in Equation (2) where the terms on the left represent the power sources to the flow with ${{\mathcal P}_S}$ the amount of shaft power related to any moving fan’s blade, and ${{\mathcal P}_{\textbf{f}}}$ the power input to the flow when the fan’s blades are substituted by a body force representation inside a volume. On the right-hand side, the sinks are represented by $W\dot h$ as the change in potential energy due to changes in altitudeFootnote ^*, ${\dot {\mathcal E}_m}$ the mechanical energy deposition rate and ${\rm{\Phi }}$ and ${\rm{\Theta }}$ (Equations (3)) the system losses manifesting themselves under the form of viscous dissipation and volumetric pressure rates, respectively.

(2) \begin{align} {{\mathcal P}_S} + {{\mathcal P}_{\textbf{f}}} &= W\dot h + \underbrace {{{\dot {\mathcal E}}_k} + {{\dot {\mathcal E}}_p} + {{\dot {\mathcal E}}_\tau }}_{{{\dot {\mathcal E}}_m}} + {\rm{\Phi }} + {\rm{\Theta }}\end{align}

(3a) \begin{align}{\rm{\Phi }} &= \mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt {\mathcal V}} {\left( {\overline {\bar \tau } \cdot \nabla } \right){\textbf{V}}{\rm{d}}{\mathcal V}} \end{align}

(3b) \begin{align}{\rm{\Theta }} &= - \mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt {\mathcal V}} {{p_{\rm{G}}}\nabla {\textbf{V}}{\rm{d}}{\mathcal V}} \end{align}

2.1.2 Flow response to nose shape

Figure 5 depicts the evolution of the BL area (Equation (1a)) both along the fuselage and within the wake, where it is observed that the hemispheric nose stands out from the other shapes. From the aircraft’s shoulder, a jump in the BL area caused by a strong local shock is noted. That shock is discernible from the mass displacement area ${{\rm{\Delta }}^{\rm{*}}}$ with a sharp increase that causes an unrecoverable thickening of the shear-layer. The semi-ellipsoid approximation of ${\rm{F}}{{\rm{R}}_{{\rm{nose}}}} = 3.71$ stands amid the other shapes that are not subject to any shock. Its deviation from both A321 profiles yields a small difference in the BL properties but suggests a quasi-identical, yet greater, thickness and energy transfer from the fuselage to the flow. In other terms, a rear-mounted BLI propulsor will experience a slightly smaller mass-flow from the elliptic approximation than the more representative contours for a given propulsor size.

Figure 5. 3D axi-symmetric boundary-layer area (top), displacement area, momentum area and kinetic energy area (left) and wake energy-flux transfers and potential for energy recovery (right) for five different nose shapes at FL350 and ${M_{\rm{B}}} = 0.82$ .

The difference in energy is confirmed by the right-hand side column’s charts that depict the flux transfer breakdown from the aircraft’s trailing edge to the Trefftz plane. The kinetic energy deposition rate ${\dot {\mathcal E}_k}$ and pressure-defect work ${\dot {\mathcal E}_p}$ are transfered into unrecoverable forms of energy ${\rm{\Phi }}$ and ${\rm{\Theta }}$ . The decay of power overtime results from the absence of a body that imparts additional energy to the flow, which self-recovers to its initial quiescent form. Whereas a steep rate of dissipation is observed on the different nose shapes, the hemispheric nose necessitates more time to fully dissipate with a more gradual rate. Notably, the volumetric pressure work ${\rm{\Theta }}$ cannot be neglected any longer as the shock left a residual irrecoverable pressure-defect work.

It is concluded that from the perspective of the BL growth, energy imparted to the flow and its recoverable part from an ideal device mounted at the aircraft’s rear, the semi-ellipsoidal nose of ${\rm{F}}{{\rm{R}}_{{\rm{nose}}}} = 3.71$ suits the nose shape approximation. However, it is stressed that the elliptic approximation is only acceptable under shock-free conditions which otherwise would dissipate a higher amount of non-recoverable energy and hence diminish the aerodynamic potential performance of the aircraft design. The elliptic nose shape is thus selected for the SMR reference aircraft and is acknowledged as a good candidate to represent the nose shape.

2.1.3 Boundary layer sensitivity to mach number

To follow on the flow response due to changes in nose geometries, a further investigation is performed using the precedent hemispheric nose for different flight Mach numbers. At a glance, a distinctive trend arises where higher free-stream Mach numbers favour shock generation and thicken the BL. For flight conditions below transonic, i.e. M0.75, no shock is reported by the BL displacement area ${{\rm{\Delta }}^{\rm{*}}}$ . Whereas M0.65 and M0.70 flight conditions do not differ in the different charts of Fig. 6, at M0.75 the flow experiences a thicker BL from the aircraft shoulder suggesting a higher amount of energy imparted, not necessarily recoverable. The growth along the fuselage does not recover and suggests higher potentials from the momentum and energy defects. Above those conditions, a shock develops at the shoulder which eventually impacts its flow downstream and intensifies with respect to flight conditions. Nevertheless, this additional defect generated does not echo with a higher aerodynamic behaviour as can be explained by the kinetic energy deposition rate downstream of the body and its transfer into viscous dissipation rate. Not only the dissipation rate increases irrecoverably but the volumetric pressure work rises as the shocks and compressibility effects become more important. In practice, the overall power increase and the portion of pressure-defect work takes over the viscous dissipation rate due to the stronger shock.

Figure 6. 3D axi-symmetric boundary-layer area (top), displacement area, momentum area and kinetic energy area (left) and wake energy-flux transfers and potential for energy recovery (right) for Mach numbers between 0.65 and 0.82 on the hemispherical nose approximation.

2.2.1 Rationale

Previous studies assessed the influence of the flight conditions on the energy recovery [Reference Lamprakis, Sanders and Laskaridis16, Reference Mutangara, Sanders, Laskaridis, Hart and Schmitz17]. More favourable effects are observed with the chord Reynolds number, body Mach number, angle-of-attack and body fineness ratio. For instance, Airbus Beluga-like aircraft suggest larger potential for energy recovery than slender bodies like the Airbus A380 and A320 families [Reference Lamprakis, Sanders and Laskaridis16], but slender bodies retain quasi zero-pressure gradient dissipative characteristics over their surfaces and allow their approximations with flat plate theory. Despite investigations on the fuselage slenderness ratio, the influence of the tail fineness ratio on the BL properties and the potential energy recuperation has not been studied, and important considerations must be taken on the quality of the incoming airflow. It is preempted that the presence of the nacelle, when integrating a propulsor around the fuselage tail, impacts the flow before it passes through the fan. Therefore, the upstream flow needs to be further analysed as it is expected to highly impact the performance of the fan (with distortion and energy harvesting). On conventional aircraft, the rear-end of the aircraft is assimilated as a non-axysimmetric frustum (to prevent tail strike) whose trailing edge serves as an exhaust to the auxiliary power unit (APU). However, for the sake of simplification in this work and to focus on BLI aerodynamics, the requirements of tail-strike up-sweep and APU have been omitted, and the tail approximated with an axi-symmetric cone.

2.2.2 Morphing sequence

First, different tail fineness ratios are defined on the conic tail. To maintain the aircraft’s chord length, the mid-body fuselage length is extended accordingly. Whilst a retrofit integration of the BLI propulsor around the conic tail can be performed [Reference Sanders22], more tightly integrated propulsion systems necessitate a morphing of the tail to axially direct the flow through the fan. In doing so, some of the integration effects can be quantified by means of comparison with a bare morphed fuselage. This is a necessary stepping stone to decompose the installation effects, as to whether they are attributed to the morphing or nacelle inclusion.

To that end, the conical tail is used as reference whereby a serpentine substitutes the initial arc leading to the cone. This serpentine offers a natural extended pre-diffusion of the flow to better direct the flow towards the fan. Ensuring adequate spacing for the propulsor’s nacelle integration, the plug-cone features a shorter length with a steeper slope. An illustrative example is given in Fig. 7 following the morphing described.

Figure 7. Tail morphing sequence from its reference (– –) to its morphed shape (—), from Ref. [Reference Moirou12].

Subsequently, Fig. 8 depicts a set of four conic tail designs (solid curves) and a morphed tail (dashed curve). The tail fineness ratio, ${\rm{F}}{{\rm{R}}_{{\rm{tail}}}}$ , is swept for a constant fuselage radius where the different tail lengths are reported from the shortest L1 to the longest L4. The closest approximation to the reference aircraft’s frustum is given by L2 which morphed tail equivalent is reported as L2’ in the following.

Figure 8. 2D axi-symmetric modelling of conic tail-cones of different lengths, and a morphed tail (dashed) derived from L2.

2.2.3 Figures of merit

In scenarios where no external force is given by either ${{\mathcal P}_S}$ or ${{\mathcal P}_{\textbf{f}}}$ , the fuselage drag power is deposited into the flow and transfered in different power forms as given in Equation (4).

(4) \begin{align} - {{\mathcal F}_{{\rm{fus}}}} \cdot {{\textbf{V}}_{\rm{B}}} = {\dot {\mathcal E}_m} + {\rm{\Phi }} + {\rm{\Theta }}\end{align}

However, part of the mechanical energy-rate present in the BL is not entirely exploitable. To segment the recoverable part, the velocity at any point of the flow-field is decomposed into its isentropic and non-isentropic components [Reference Aguirre, Duplaa, Carbonneau and Turnbull23]. Therefore, applying the decomposition in Equation (5), $\dot {\mathcal E}_m^{\rm{*}}$ denotes the isentropic portion and $\overline {{{\dot {\mathcal E}}_m}} $ the non-isentropic portion of the mechanical energy.

(5) \begin{align}{\dot {\mathcal E}_m} = \dot {\mathcal E}_m^{\rm{*}} + \overline {{{\dot {\mathcal E}}_m}} \end{align}

Introduced by Sanders and Laskaridis, the potential for energy recovery (PER) is used as a means of describing the portion of the BL flow rich in non-dissipated energy potentially extractable by a mechanical device. Initially defined in Ref. [Reference Sanders and Laskaridis4] with the viscous dissipation rate at the trailing edge of the bare fuselage ${{\rm{\Phi }}_{{\rm{TE}}}}$ (Equation (6a)), the definition is later corrected in Ref. [Reference Lamprakis, Sanders and Laskaridis24] and approximated to account for the irreversible compressibility-induced losses ${{\rm{\Theta }}_\infty }$ (Equation (6b)). These two definitions are volumetric integrals and do not enable the accumulation of PER to be tracked along the surface. In Ref. [Reference Mutangara, Sanders, Laskaridis, Hart and Schmitz17], the accumulation of the non recoverable energy $\overline {{{\dot {\mathcal E}}_m}} $ relative to the overall energy-rate deposited by the fuselage can be quantified at any given survey plane along a surface or in the wake (Equation (6c)). Using the velocity decomposition method, it is now possible to apply the strict definition of PER whereby the irreversible compressibility-induced losses ${{\rm{\Theta }}_{{\rm{irr}},{\rm{TE}}}}$ are not approximated with their residual value (Equation (6d)). In essence, ${\rm{\Theta }}$ is split in its reversible and irreversible components from a power balance re-arrangement, and the ideal maximal energy recovery is quantified.

(6a) \begin{align}{\rm{PE}}{{\rm{R}}_{{\rm{Sanders}}}} &= 1 - \frac{{{{\rm{\Phi }}_{{\rm{TE}}}}}}{{{{\mathcal F}_{\rm{B}}}{V_{\rm{B}}}}}\\[-5pt]\nonumber\end{align}

(6b) \begin{align}{\rm{PE}}{{\rm{R}}_{{\rm{Lamprakis}}}} &= 1 - \frac{{{{\rm{\Phi }}_{{\rm{TE}}}} + {{\rm{\Theta }}_\infty }}}{{{\mathcal F}{V_{\rm{B}}}}}\\[-5pt]\nonumber\end{align}

(6c) \begin{align}{\rm{PE}}{{\rm{R}}_{{\rm{Mutangara}}}} &= \frac{{\overline {{{\dot {\mathcal E}}_m}} }}{{{{\mathcal F}_{\rm{B}}}{V_{\rm{B}}}}}\\[-5pt]\nonumber\end{align}

(6d) \begin{align}{\rm{PER}} &= 1 - \frac{{{{({\rm{\Phi }} + {{\rm{\Theta }}_{{\rm{irr}}}})}_{{\rm{TE}}}}}}{{{{\mathcal F}_{\rm{B}}}{V_{\rm{B}}}}}\end{align}

2.2.4 Aerodynamic sensitivity

To characterise the flow sensitivity to changes in the plug-cone shape, Fig. 9 depicts the BL area and its kinetic energy area along with ${\rm{PE}}{{\rm{R}}_{{\rm{Mutangara}}}}$ for all conic tails and its morphed derivative. It appears that the longer the fuselage mid-section (i.e. the shorter the tail), the slightly thicker the BL and its excess properties along the mid-body as the wetted area increases and more energy is deposited into the flow. These discrepancies are nonetheless quite negligible at the trailing edge but play an important role as the flow experiences different pressure gradients around the tail.

Figure 9. 3D axi-symmetric boundary-layer area (top), kinetic energy area (left) and potential for energy recovery from Equation (6c) (right) for the four conic tails of different lengths and a morphed derivative at FL350 and ${M_{\rm{B}}} = 0.82$ .

First, as the BL flow approaches the tail (represented by the vertical dashed lines for the different tails), it is dragged down to follow the body’s curvature which locally accelerates before diffusing and decelerating along the slope as it experiences an adverse pressure gradient. For instance, L1 experiences a steeper boat angle than L4 causing a sharper rise in velocity around the convex fuselage region therefore contributing towards a local defect in energy as the velocity at any point within the BL is larger than its edge quantity.

The comparison of the conic tail L2 with its morphed derivative L2’ follows in the last row of Fig. 9 whereby an important drop in BL properties is experienced before a sharp bounce back. This phenomenon is justified by the steeper change in geometry curvature which then plateaus – to later integrate the propulsor, and then a replica of the precedent curvature at the plug-cone. This double phenomenon in BL properties results from two strong favourable pressure gradients intertwined by adverse gradients. However, the last drop in ${{\rm{\Theta }}^{\rm{*}}}$ is smaller in amplitude than the previous as the flow pre-diffused. It is opined that the consideration of either cone shape has an impact on the BL properties at any location along the tail and results in a benefit in excess mechanical energy that can be harvested by a device.

Tracking the accumulation of the non recoverable energy relative to the overall energy-rate deposited by the fuselage, PER is quantified at any given survey plane along the wall surface, and in the wake. Nonetheless, despite allowing one to track the accumulation of energy deposition rate that can be harvested mechanically, and being a contribution to this paper, the progress of PER along the body is subject to several factors. Its comparison with the definitions from previous works [Reference Sanders and Laskaridis4, Reference Lamprakis, Sanders and Laskaridis24], given in Fig. 10, yields significant discrepancies. Notably, larger differences for L1 than for L4 are explained by shorter and steeper boat tails which cause adverse pressure gradients and affect the reported accumulation of $\overline {{{\dot {\mathcal E}}_m}} $ as seen with Fig. 9. These pressure gradients are responsible for large deviations given in ${\rm{\Theta }}$ and predominantly in its reversible component ${{\rm{\Theta }}_{{\rm{rev}}}}$ . This sharp rise in ${\rm{\Theta }}$ is the result of a larger isentropic compression and dilation process captured by the other definitions reported in Fig. 10. Additionally, a delay is noted between the kinetic energy area ${{\rm{\Theta }}^{\rm{*}}}$ and PER. This is caused by the difference in survey plane definitions. In the former, planes perpendicular to the wall surface are used to the BL edge to determine the BL properties whereas vertical surveys are defined in the decomposition method to sweep over the entire domain height. Lastly, the fineness of the meshes generated alter the decomposition yielding higher numerical dissipation. As the tail length increases, slender aft-bodies experience lower pressure gradients but generate viscous dissipation as a result of shear stresses along the wall surface. Despite ensuring fine meshes to capture the viscous sub-layer, this turbulent region would need very fine grids to decompose well the velocity components. However, it is found that the viscous dissipation rate is highly sensitive to the mesh fineness and non-dimensional first cell heights ${y^ + }$ below one underestimate its value.

Figure 10. Comparison of the potential for energy recovery refinement definitions on the different tails from Figure 8 at the fuselage trailing edge, with the PER value of the present study (Equation (6d)) on the tail L2 given in a dashed line for ease of comparison.

Overall, a decreasing trend in PER values is noted with respect to the tail length. The comparison of the conic tail with its morphed derivative yields a larger recovery potential as postulated. The concave diffusive slope imparts more energy to the flow (thanks to a larger fuselage drag power) which does not dissipate upstream of the survey plane and reports larger mechanical benefits.

Morphed geometries locally favour pressure-defect work over viscous dissipation, contributing to larger PER (Fig. 10). This can be explained by the increased dilation from changes in curvatures and the larger wetted area. Nonetheless, all the energy is not transformed into ${\rm{\Theta }}$ or ${\rm{\Phi }}$ leaving a greater aerodynamic potential as given by PER. Importantly, it is reminded that greater potentials do not necessarily translate to greater benefits. Figure 11 depicts that transfer of energy between L2 and L2’ whereby slightly smaller diffusive effects are observed for the morphed tail region. It appears however clear that at the fuselage trailing edge, the non recoverable energy $\overline {{{\dot {\mathcal E}}_m}} $ is slightly under-predicted, as previously mentioned, and therefore negatively impacts ${\rm{PE}}{{\rm{R}}_{{\rm{Mutangara}}}}$ value, which can still be used as a conservative approach. With the integration of a propulsor which fan could be placed around 50–60% of the tail length (Fig. 9), ingesting the bottom layer of the BL rich in energy would reduce the fan power requirement to operate as well as reduce the integration penalties, e.g. over-cowl viscous dissipation. However, as expressed before, the pressure gradients faced by the flow locally affect PER around the mid tail to a larger extent with the conic than the morphed tail. Despite advising to integrate a thruster around the middle of the morphed tail, the morphology plays an important role in the recoverable work and the fan size still needs to be assessed.

2.3.1 Slope angle effects

Unlike the previous investigation on the conic tails with their derivation in a morphed candidate, the flight speed is reduced to ${M_{\rm{B}}} = 0.78$ ( $Re = {2.73.10^8}$ ) for the rest of the paper to agree with the cruise mission of the PFC and determine a morphed baseline at its condition. To understand the influence of the shaping, morphed fuselages are designed with different combinations of slope lengths and angles. First, the pre-diffusive slope is investigated to help direct the flow before entering the propulsor. As such, the slope angle is defined as ${\psi _{slope}} = {L_{{\rm{slope}}}}/\left( {{R_{{\rm{fus}}}} - {R_{{\rm{hub}}}}} \right)$ between the tail starting point at the fuselage radius and the hub radius of the envisaged propulsor throat position, marking the end of the slope.

Figure 11. Energy decomposition and their evolution along the fuselage surface and in the mid-field wake.

Each fictitious position of the propulsor’s throat is discretised in 40 survey planes equi-spaced from the fuselage wall to the BL edge, where the mass-flow averaged Mach number and mass-flow rate are computed. Figure 12 reports the Mach number relative to the fan-face target to ensure its operation with respect to all morphed tail slope angles ${\psi _{slope}}$ to depict the influence of the pre-diffusive slope. The cloud bounds the Mach number extrema for target ingestions of 20–100% of the BL mass-flow, following a kernel distribution. Inside it, the mid-height scatter quantifies the mean Mach number of this design, and its surrounding box covers the 25 $^{{\rm{th}}}$ to 75 $^{{\rm{th}}}$ quartiles which extend with matches representing the extrema (not including outliers). Additionally, a filter is applied on the mass-flow rate integrals to ensure that no reverse flow is captured at the axial position which would enlarge the throat area to capture the target mass-flow rate and increase the load and distortion faced by the fan.

Figure 12. Distribution of mass-flow averaged Mach numbers at the slopes’ end-points with respect to different inclination angles, with 3 identified designs for later use, relative to a threshold of 90% of the Mach number target.

For ${\psi _{slope}}$ larger than 13 ${{\rm{\;}}^ \circ }$ in Fig. 12, more than 75% of the calculated Mach numbers on the survey planes fall below the 90% threshold defined. To ensure the condition is met at the fan on these designs, its size should be large enough to ingest enough mass-flow, and its intake should undesirably be designed convergent to locally accelerate the flow generating larger total pressure distortion. Therefore, morphed fuselages which slope angles causing Mach numbers to fall under the threshold as well as Mach numbers that will lead to only large propulsors are discarded, and their integrated propulsive concepts not designed.

Conversely, for designs leading to Mach numbers close to the target, propulsor integration effects will naturally slow down the flow before crossing the intake towards the fan. As for the designs with Mach numbers above the target, additional diffusion is ensured within the propulsor intake to reduce the flow speed.

2.3.2 Plug angle constraints

In addition to the constraints on the slope angle, the integration of the propulsor along the tail directly constrains the plug length (for a given tail length) as well as its angle (with respect to the slope angle). As such, if the change in curvature is too important, the flow separates downstream of the fuselage impacting its aerodynamic performance. Through the integration of the retrofit propulsor around the morphed tail, the flow exiting the exhaust at a higher speed than the bare fuselage will also likely separate. To prevent the generation of these propulsive designs, the pressure and skin-friction coefficients are tracked along the plug surface. The absence of friction on a body’s surface suggests a flow separation, and as a by-product a sharp rise in the pressure coefficient. To illustrate, the flow-field around the tail’s plug is reported in Fig. 13 for the three designs previously identified in Fig. 12.

Figure 13. Computed Mach number contours around the three designs identified in Fig. 12.

As a result of separation on Fig. 13(a) for ${\psi _{slope}} = {19.8^ \circ }$ , the Mach numbers and mass-flow rates computed for Fig. 12 suggested reversed flow on the survey planes which disregarded the designs previously identified. Conversely, for Fig. 13(c) with ${\psi _{slope}} = {6.6^ \circ }$ , a large majority of the flow conditions computed fell above the fan requirements, but could be balanced with a well-design propulsor intake. On these designs, the plug length is constrained (as the tail length is imposed) featuring a steep angle of ${\beta _{plug}} = {33.3^ \circ }$ , showcasing separation downstream of the fuselage. Following the fair assumption that separation will remain for powered aircraft, fuselages like design C are also disregarded. Consequently, when both slope and plug angles fall below their critical angles, flow-fields similar to the design B in Fig. 13(b) are obtained.

Retrieving Fig. 12, the design space is bounded by the critical slope angle and its induced plug angle between 7 and 13 ${{\rm{\;}}^ \circ }$ , with steep plugs and steep slopes, respectively. These designs are represented in Fig. 14 where designs like A are removed from the bottom-left corner, and like C from the top-right corner (missing scatters).

Figure 14. Schematic of two morphed fuselages for ${L_{{\rm{slope}}}}/{L_{{\rm{tail}}}} = 0.4$ and $0.5$ , and s ${R_{{\rm{huba}}}}/{R_{{\rm{fus}}}} = 0.6$ and $0.4$ , respectively, in grey and yellow with $\bigcirc$ .

2.3.3 Slope length and hub radius effects

Further to the narrowed-down morphed fuselages and from the possibility of tracking PER along the fuselage, the definition from Mutangara et al. [Reference Mutangara, Sanders, Laskaridis, Hart and Schmitz17] is followed. Based on the designs given in Fig. 14 with respect to different hub radii and slope lengths (in markers), ${\rm{PE}}{{\rm{R}}_{{\rm{Mutangara}}}}$ values are reported in Fig. 15.

Figure 15. Potential for energy recovery with respect to the different slope lengths and hub radii given in Fig. 14.

As expressed above, more aggressive slope designs provide additional energy to the viscous flow, locally present in the form of pressure, with low shear stresses hence contributing to higher PER values as viscous dissipation is delayed. This phenomenon is observed where the slope length is constrained to low values featuring a somewhat sharp ramp. With increasing slope lengths, PER decreases as more streamlined flows experience larger shear stresses responsible for viscous dissipation. For the same reasoning, low hub radii favour PER as the rise in energy deposition rate is not yet converted into irrecoverable losses. In addition, the monotonicity in PER with hub radii is respected regardless of the slope length.

Lastly, the uneven distribution for ${L_{{\rm{slope}}}}/{L_{{\rm{tail}}}} = 0.4$ and $0.6$ is observed as the lowest hub radius of the former axial location does not experience reverse flow but is still subject to a strong pressure gradient contributing to ${\dot {\mathcal E}_p}$ locally, increasing the perceived PER. Conversely for the latter location, the opposite is observed for higher radii where the somewhat steep plug angle affects the mechanical energy-rate terms at the survey station and reduces the potential for energy recovery.

3.3.1 Power consumption and savings

In this work, retrofit state-of-the-art turbofans designed by Zhao and Yao [Reference Zhao and Yao27] and comparable to the CFM LEAP-1 or PW1100G are used. Therefore, all PSC values reported under-estimate the real savings the novel aircraft might achieve as in real case scenarios, the main turbofans should be down-scaled as less thrust is required from them. Notably, there would be additional savings on nacelle drag and weight (albeit the latter is not considered here). Nevertheless, this approach allows one to compare all propulsor designs generated in a similar manner by only assessing the benefits from the thruster and not considering any further aero-propulsive benefits from the re-scaling.

To that end, the shaft power ${{\mathcal P}_S}$ from the power balance in Equation (2) describing the aft-mounted fan power demand is given as ${{\mathcal P}_{{\rm{BLI}}}}$ to differentiate it from the main turbofans ${{\mathcal P}_{{\rm{pod}}}}$ :

(7) \begin{align}{{\mathcal P}_{{\rm{BLI}}}} = \mathop \int\!\!\!\int \nolimits_{{{\mathcal S}_{\rm{P}}}} \left[ {\rho \left( {e + \frac{{{{\textbf{V}}^2}}}{2}} \right){\textbf{V}} + p{\textbf{V}} - \overline {\bar \tau } \cdot {\textbf{V}}} \right] \cdot \hat n{\rm{d}}{\mathcal S}\end{align}

In the turbo-electric aircraft architecture considered, this energy is an electrical power demand which needs to be first converted from mechanical energy, as extracted from the low-pressure turbine (LPT) of the under-wing turbofans. However, this modelling is out of the scope of this paper and rather, a transmission efficiency is introduced as ${\eta _{trans}}$ . The actual power extracted from the low-pressure shaft is given in Equation (8) and related to the overall power consumption ${{\mathcal P}_{{\rm{PFC}}}}$ in Equation (9):

(8) \begin{align}{{\mathcal P}_{{\rm{BLI}},{\rm{PFC}}}} &= \frac{{{{\mathcal P}_{{\rm{BLI}}}}}}{{{\eta _{trans}}}}\end{align}

(9) \begin{align}{{\mathcal P}_{{\rm{PFC}}}} &= {{\mathcal P}_{{\rm{pod}},{\rm{PFC}}}} + {{\mathcal P}_{{\rm{BLI}},{\rm{PFC}}}}\end{align}

whereby ${{\mathcal P}_{{\rm{pod}},{\rm{PFC}}}}$ accounts for two turbofans mounted on either sides.

Relative to the reference aircraft that only produces the required forward force, the power saving coefficient is defined with:

(10) \begin{align}{\rm{PSC}} = 1 - \frac{{{{\mathcal P}_{{\rm{PFC}}}}}}{{{{\mathcal P}_{{\rm{A}}/{\rm{C}},{\rm{ref}}}}}} = 1 - \frac{{{{\dot m}_{f,{\rm{PFC}}}}}}{{{{\dot m}_{f,{\rm{A}}/{\rm{C}},{\rm{ref}}}}}}\end{align}

which is reducible to the right-hand side formulation with ${\dot m_f}$ denoting the fuel-flow rate, assuming that the same fuel properties are considered in both aircraft.

In this work, the latter formulation is used based on engine decks for combinations of power off-takes and propulsive force generation. To that end, the fuel saving coefficient (FSC) is used in lieu of the PSC but represents the same metric.

3.3.2 Propulsive force share

In the PFC, the propulsive force is split in between the under-wing engines and the aft-mounted thruster. The portion of force generated from the latter relative to the main turbofans in given by the propulsive force share (PFS):

(11) \begin{align}{\rm{PFS}} = \frac{{{{\mathcal F}_{{\rm{BLI}},{\rm{PFC}}}}}}{{{{\mathcal F}_{{\rm{PFC}}}}}}\end{align}

This non-dimensional form shows its importance when one designs different aircraft and evaluates the change in propulsive force under equivalent power supply, or inversely, the change in power requirement to maintain the same force. The overall force requirement from the PFC ( ${{\mathcal F}_{{\rm{PFC}}}}$ ) is estimated from a drag build-up isolating the propulsive fuselage from the rest of the aircraft [Reference Moirou12]. All forces ${\mathcal F}$ are computed from near-field integration of pressure and friction around wall surfaces:

(12) \begin{align}{\mathcal F} = \mathop \int\!\!\!\int \nolimits_{\mathcal S} \left( {p - {p_\infty }} \right)\hat n - \overline {\bar \tau } \cdot \hat n{\rm{d}}{\mathcal S}\end{align}

To obtain the direct contribution of the propulsive fuselage, the forces ${{\mathcal F}_{{\rm{fus}} + {\rm{P}},{\rm{PFC}}}}$ on the propulsive fuselage are compared against the morphed clean fuselage ${{\mathcal F}_{{\rm{fus}},{\rm{PFC}}}}$ previously used by associating in change in force distribution to the propulsor integration:

(13) \begin{align}{{\mathcal F}_{{\rm{BLI}},{\rm{PFC}}}} = {{\mathcal F}_{{\rm{fus}} + {\rm{P}},{\rm{PFC}}}} - {{\mathcal F}_{{\rm{fus}},{\rm{PFC}}}}\end{align}

4.3.1 Intake length reduction

Larger distortion is experienced by a thruster fan than with a clean free-stream flow ingested by a turbofan mounted under the wings [Reference Kenway and Kiris15]. Despite the larger distortion, the pre-diffusion ensures a homogeneous circumferential flow distribution to the throat and only the radial distribution (span-wise) is investigated. In events where the pre-diffusion is not sufficient for the flow to reach the target Mach number at the fan-face, further diffusion is obtained from the intake design. With the consideration of compact nacelles around the propulsors, the total pressure distribution and radial distortion index (RDI) [Reference Zachos, Frascella, MacManus and Gil-Prieto28] are studied as complex flow diffusion from the intake and acceleration from the fan are combined.

Both propulsor designs from Fig. 18 are investigated whereby 25% and 100% of the BL mass-flow are ingested. Short intakes are considered where the length over diameter ratio (L/D) is reduced from 1 to 0.7 and 0.4, respectively. In comparison, state-of-the-art under-wing engines’ intakes range between 0.65 and 0.85 for SMR aircraft [Reference Malsaac and Roy29] whereas short intakes are designed around 0.5 [Reference Peters, Spakovszky, Lord and Rose30].

Figure 22 depicts the distribution of the total pressure at the fan-face normalised by the free-stream total pressure for two propulsor sizes under the different intake lengths and fan pressure ratios. Similarly for both propulsor throat sizes, an increase in intake compactness sees the AIP pushed further forward (i.e. closer to the throat) increasing the total pressure distribution close to the nacelle shroud. Notably on the larger propulsor, suction effects in the lower region caused by larger fan pressure ratios reduce the total pressure distribution as the flow is slightly accelerated. Closer to the tip, the same phenomena are observed on all propulsors whereby larger diffusion, i.e. total pressure, occurs for longer intakes but the BL forming on the interior of the nacelle reduces the total pressure distribution.

Figure 22. Normalised distribution of the total pressure along the fan span for the small propulsor $R1$ (left) and large propulsor $R2$ (right) under two fan pressure ratio and three intake lengths.

As regards the RDI, values are reported in Table 1. For the different propulsors considered, increases between 10 and 14% in RDI are observed when the intake L/D is reduced from 1.0 to 0.4. Although these increases are non-negligible and at the fan designer’s discretion [Reference Castillo Pardo and Hall31], in this study the fan is modelled with ideal boundary conditions and therefore not transporting the experienced distortion through the fan stage which would be expected to increase the viscous dissipative losses. As degrees of fidelity are increased in the fan modelling, one should not omit the distortion effects and adjust the compactness of the intake to mitigate them.

4.3.2 Plug-cone extension

From the geometries studied above and their intake length reductions, the nacelle lengths are reduced by the same amount to maintain an equivalent exhaust than the baseline. Consequently, this elongates the plug-cone and reduces its angle. For small propulsors, the reduction in nacelle length does not impact considerably the plug shape and its effects on the surrounding flow-field. However, with larger propulsors, the elongation offered in the plug-cone prevents abrupt curvature changes with smaller plug angles and favours the force distribution.

Notably, Fig. 23 depicts the large propulsor operating at two fan pressure ratios for the baseline and compact nacelles. Shrinking the propulsor’s nacelle enables the plug-cone length to be extended and its angle to be reduced from 35.8 to 25 ${{\rm{\;}}^ \circ }$ . The reduction in angle yields a smaller entropy generation behind the cone but not necessarily a better force redistribution on the plug arc and cone. As the jet velocity increases with FPR, the flow is more unlikely to remain attached to the wall surface and reducing the plug angle positively contributes to the net axial force. Notably, at low FPR, the reduction in plug angle from 35.8 to 25 ${{\rm{\;}}^ \circ }$ favours the force redistribution along the plug but its benefit is outbalanced by the local distribution within the exhaust. As the FPR increases, the benefits offered by a smoother plug arc contribute to beneficial forward forces on the rear-ward facing surfaces whilst maintaining equivalent forces within the exhaust. These observations lead to slender plug-cone shapes preventing sharp accelerations along the plug arc just downstream the exhaust and contributing to large entropy losses.

Table 1. Radial distortion indices for two propulsor’s intake dimensions and two fan pressure ratios

Figure 23. Contours of entropy around large propulsors ( $R2$ ) ingesting the entire boundary layer under fan pressure ratios of (a) 1.20 and (b) 1.50 for two intake/nacelle/plug-cone length combinations.

4.3.3 Overall benefits

In addition to the benefits obtained on the force redistribution along the plug-cone, improved aerodynamic performance arises from the reduction in size of the nacelle length with a smaller wetted surface and a slight reduction in entropy, as observed in Fig. 23. From an energy standpoint, using a compact nacelle for the propulsor over the baseline reduces the overall viscous dissipation rate ${\rm{\Phi }}$ by 4 and 20% for the larger propulsor throat area, under FPR ratios of 1.20 and 1.50, respectively. These benefits therefore translate to lower fan power demand for the same net axial force, thus power savings, or to a larger force contribution from the thruster relative to the two main engines under the wings.

Nonetheless, these observations are made on a unique propulsor position along the tail and cannot be generalised to all combinations shown in Fig. 17. Whereby possible, the intake length is therefore reduced to favour more compact nacelles, leaving a longer and less steep plug-cone, to offer benefits to the flow redistribution and energy utilisation. An example of optimisation studies on compact exhausts is studied in Ref. [Reference Matesanz-García, Piovesan and MacManus32] whereby improvements of 0.32% of the total aircraft thrust requirement is observed from their reference.