2.1 Nacelle parametrisation

The parametric definition of the nacelle geometry is controlled with an analytical approach based on the intuitive class shape transformation method (iCST) method [Reference Heidebrecht and MacManus24]. It follows Kulfan’s CST approach [Reference Kulfan27] and has been extended to use intuitive design variables. For this study, the nacelle shape is controlled with seven intuitive parameters: ${r_{hi}}$ , ${L_{nac}}$ , ${r_{te}}$ , ${r_{if}}$ , ${f_{{\rm{max}}}}$ , ${r_{{\rm{max}}}}$ and ${\beta _{nac}}$ (Fig. 2). These geometric variables may be constrained by the requirements of the intake and exhaust systems as well as integration and ground clearance considerations. For example, the nacelle trailing edge ( ${r_{te}}$ ) is usually defined by the exhaust flow requirements and the minimum nacelle boat-tail angle ( ${\beta _{nac}}$ ) is set by the external aero-line of the bypass duct. In addition, the nacelle maximum radius ( ${r_{{\rm{max}}}}$ ) plays an important role to accommodate prescribed keep-out-zones for auxiliary components.

Figure 2. Intuitive nacelle design variables.

The general form of a CST function curve ( $\xi \left( \psi \right)$ ) can be described as Equation (1):

(1) \begin{align}\xi \left( \psi \right) = C_{{N_2}}^{{N_1}}\left( \psi \right)S\left( \psi \right) + \psi {\rm{\Delta }}{\xi _{te}};{\rm{\;\;\;}}\xi = \frac{z}{c},{\rm{\;\;\;}}\psi = \frac{x}{c}\end{align}

where $C_{{N_2}}^{{N_1}}\left( \psi \right)$ is the class function, $S\left( \psi \right)$ is the shape function, ${\rm{\Delta }}{\xi _{te}}$ defines $z$ offset between the curve endpoints, $c$ is the curve length in the x-direction, and $x$ and $z$ are the axial and radial absolute coordinates in the Cartesian space [Reference Kulfan27].

The round-nosed, sharp trailing-edge aerofoil class function is used for the parametric definition of the nacelle. This corresponds to values of ${N_1}$ = 0.5 and ${N_2}$ = 1 for the definition of the class function (Equation (2)):

(2) \begin{align}C_{1.0}^{0.5}\left( \psi \right) = {\psi ^{0.5}}\left( {1 - \psi } \right)\quad {\rm{for}}\quad 0 \le \psi \le 1\end{align}

The shape function is defined as follows (Equation (3)):

(3) \begin{align}S\left( \psi \right) = \mathop {\mathop \sum \limits_{i = 0} }\limits^n b{p_i}\Big( {\begin{array}{*{20}{l}}n\\[-2pt] i\end{array}} \Big){\psi ^i}{(1 - \psi )^{n - i}}\end{align}

where $\Big( {\begin{array}{*{20}{l}}n\\[-2pt] i\end{array}} \Big)$ denotes the binomial coefficients and $b{p_i}$ are the different coefficients of the Bernstein polynomials.

The general form for $\xi \left( \psi \right)$ and its first derivative can be expressed as Equations (4) and (5):

(4) \begin{align}\xi \left( \psi \right) = {\psi ^{0.5}}\left( {1 - \psi } \right)\mathop {\mathop \sum \limits_{i = 0} }\limits^n b{p_i}\Big( {\begin{array}{*{20}{l}}n\\[-2pt] i\end{array}} \Big){\psi ^i}{(1 - \psi )^{n - i}} + \psi {\rm{\Delta }}{\xi _{te}}\end{align}

(5) \begin{align}\begin{array}{*{20}{l}}{\frac{{\partial \xi \left( \psi \right)}}{{\partial \psi }} = 0.5{\psi ^{ - 0.5}}\left( {1 - \psi } \right)\mathop {\mathop \sum \limits_{i = 0} }\limits^n b{p_i}\Big( {\begin{array}{*{20}{l}}n\\[-2pt] i\end{array}} \Big){\psi ^i}{{(1 - \psi )}^{n - i}} - {\psi ^{0.5}}\mathop {\mathop \sum \limits_{i = 0} }\limits^n b{p_i}\Big( {\begin{array}{*{20}{l}}n\\[-2pt] i\end{array}} \Big){\psi ^i}{{(1 - \psi )}^{n - i}} + }\\[15pt] { + {\psi ^{0.5}}\left( {1 - \psi } \right)\mathop {\mathop \sum \limits_{i = 0} }\limits^n b{p_i}\Big( {\begin{array}{*{20}{l}}n\\[-2pt] i\end{array}} \Big){H_{n,r}}\left( \psi \right) + {\rm{\Delta }}{\xi _{te}}}\end{array}\end{align}

where ${H_{n,r}}\left( \psi \right)$ is expressed as Equation (6):

(6) \begin{align}{H_{n,r}}\left( \psi \right) = i\left( {{\psi ^{i - 1}}} \right){(1 - \psi )^{n - i}} - {\psi ^i}\left( {n - i} \right){(1 - \psi )^{n - i - 1}}\end{align}

This mathematical derivation can be extended to any n-order derivation and obtain an analytical expression. For this specific application in which the class shape is defined with $C_{1.0}^{0.5}$ , the first and last Bernstein polynomial coefficients can be analytically calculated with Equation (7):

(7) \begin{align}b{p_0} = {\left( {\frac{{2{R_{le}}}}{c}} \right)^{0.5}}{\rm{\;\;\;}};{\rm{\;\;\;}}b{p_n} = {\rm{tan}}\left( \beta \right) + \frac{{\delta {z_{te}}}}{c}\end{align}

where ${R_{le}}$ is the radius of curvature at $\psi = 0$ and ${\rm{tan}}\beta $ is the gradient at $\psi = 1$ [Reference Christie, Heidebrecht and MacManus28]. For the different intuitive variables of ${r_{hi}},{L_{nac}},{r_{te}},{r_{if}},{f_{{\rm{max}}}},{r_{{\rm{max}}}}$ and ${\beta _{nac}}$ , constraints can be imposed in the form of Equations (1) and (5) to formulate a linear set of equations and obtain the coefficient of the Bernstein polynomials ( $b{p_i}$ ). This results in a set of analytical expressions as a function of intuitive design variables.

2.2 CFD approach

The grids are automatically generated with a fully-structured multi-block domain using the commercial Ansys ICEM mesher. The first cell layer height is adjusted to have ${y^ + }$ below 1. The mesh resolution is the same as reported by Heidebrecht et al. [Reference Heidebrecht and MacManus24], and comprised approximately 40,000 cells (Fig. 3(a)). For a range of nacelle configurations at the envisaged typical cruise Mach number, it has a grid converge index [Reference Roache29] in the order of 1–1.5% based on a coarser and finer mesh with around 10,000 and 160,000 cells, respectively. The CFD simulations are conducted with the viscous and compressible implicit flow solver Ansys Fluent, with a double-precision and density-based approach. The primitive flow variables are obtained by resolving the Favre-averaged Navier-Stokes equations with the k- $\omega $ shear-stress transport (SST) turbulence model. A Green-Gauss node-based scheme with a second-order upwind spatial discretisation is used. The thermal conductivity is computed with the kinetic theory and the dynamic viscosity is calculated from Sutherland’s law. For all simulations, the convergence criteria is based on a reduction of normalised residuals of four order of magnitude and an oscillation of the axial force on the nacelle lower than 0.05% over the last 200 iterations. The farfield is modelled with pressure-farfield conditions by imposing the Mach number, and the static temperature and pressure. The fan-face is defined with a pressure-outlet condition that uses an average static pressure, which is adjusted to fulfill a prescribed target massflow. An inlet boundary condition for the exhaust is used with farfield total pressure and temperature. This creates a generic exhaust broadly independent of specific engine design and reduces jet entrainment effects [Reference Heidebrecht and MacManus24]. The intake and fan-cowl are specified with no-slip walls and the nozzle walls with slip conditions [Reference Heidebrecht and MacManus24]. An example of a CFD solution obtained with the described methodology is presented in Fig. 3(b). The CFD approach has been previously validated [Reference Robinson, MacManus and Sheaf30], with a nacelle drag uncertainty within the cruise segment of 3.5% with respect to measurements. It is important to note that the artificial neural networks in this work cannot be validated with the available experimental data. The nacelle that was measured used a cylindrical centrebody which was typical for conventional architectures. However, the proposed nacelle parametric definition for future civil aero-engine (Section 2.1) is based on a continuous curvature distribution without a cylindrical centrebody.

Figure 3. Overview of the computational approach for nacelle applications.

2.3 Data sampling and neural network generation

The aerodynamic database was formed using six intuitive geometric variables ( ${L_{nac}}$ , ${r_{te}}$ , ${r_{if}}$ , ${f_{{\rm{max}}}}$ , ${r_{{\rm{max}}}}$ and ${\beta _{nac}}$ in Fig. 2) and covered a wide design space [Reference Heidebrecht and MacManus24]. For example, the normalised nacelle length varied from ${L_{nac}}/{r_{hi}}$ = 4.5 to 2.0, which covers conventional nacelles as well as compact civil architectures that are expected for ultra-high bypass ratio engines [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31]. The aerodynamic degrees of freedom were formed by the Mach number and massflow capture, and encompassed conditions representative of the cruise-point as well as off-design [Reference Boscagli, MacManus, Tejero, Sabnis, Babinski and Sheaf18]. The large bounds enable the building of a generalised model with enough flexibility across the design space, which is usually required at an early stage of the design process.

An adequate data sampling is a key aspect for the generation of complex low order models with non-linear data. Well-established sampling methods, such as the Latin hypercube sampling (LHS) technique [Reference Helton and Davis32], may not be an optimal sampling strategy for nacelle applications. Typically, it is expected that nacelle drag gradients will be greater for changes in operating conditions, especially at high Mach numbers, compared with nacelle geometry perturbations. For this reason, a mixture of approaches has been selected in which a LHS is used for the geometric space and an anisotropic sampling [Reference Zhang and Li33] for the aerodynamic degrees of freedom. In this respect, a greater refinement of sample points is used for high Mach numbers and low MFCRs. The Prandtl–Glauert factor ( $\sqrt {1 - {M^2}} $ ) [Reference Glauert34], is used to bias the sampling density towards larger Mach numbers. For the MFCR, the sampling was selected to be linear with ${\sqrt {MFCR} ^{ - 1}}$ to increase density towards lower MFCR, due to its larger nacelle drag sensitivity [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31]. The upper bound for the massflow capture ratio was set to a 95% of the MFCR at which the intake would choke using an isentropic process. Conversely, the lower bound was set to MFCR = 0.5, which could be encountered at off-design conditions [Reference Boscagli, MacManus, Tejero, Sabnis, Babinski and Sheaf18].

The ANN is built in the Python 3.7 implementation of Keras with Tensorflow as backend [Reference Abadi, Barham, Chen, Chen, Davis and Dean35]. The ANN is composed by eight neurons as the input layer, which encompass six geometry degrees of freedom ( ${L_{nac}}$ , ${r_{te}}$ , ${r_{if}}$ , ${f_{{\rm{max}}}}$ , ${r_{{\rm{max}}}}$ and ${\beta _{nac}}$ ) and two aerodynamic degrees of freedom (M and MFCR). The output layer has a single neuron that represents the nacelle drag. Both layers, i.e. input and output, are connected by a series of hidden layers. There are no general guidelines to build an ANN because it is case dependent and aspects such as the dimensionality of the problem, the number of samples or the non-linearity of the output will affect the overall neural network architecture [Reference Bouhlel, He and Martins11]. As such, the optimal definition of hyperparameters like the number of layers, number of neurons per layer or activation function are problem-dependent. Different hyperparameters were used to fine tune the predicted accuracy of the ANNs (Table 1).

Table 1. Neural network hyperparameters considered

Within the context of neural network architectures, the output of each neuron ( ${y_k}$ ) is obtained by evaluating an activation function ( $\varphi $ ) with the sum of the weighted inputs to the neuron ( ${u_k}$ ) and a bias factor ( ${b_k}$ ), Equation (8):

(8) \begin{align}{y_k} = \varphi \left( {{v_k}} \right) = \varphi \left( {{u_k} + {b_k}} \right)\end{align}

As part of this work, three activation functions were considered (Table 1). The rectified linear unit (ReLu) activation function (Equation (9)) is commonly used for artificial neural networks due to its simplicity [Reference Goodfellow, Bengio and Courville36]. The main limitation is that its gradient is equal to 0 for ${v_k} \lt 0$ [Reference Lau and Lim37], which can prevent learning because the neuron is not activated during the back-propagation process [Reference Hannun, Maas and Ng38]. This activation function has been successfully demonstrated for nacelle applications in a narrow design space with modest changes in the nacelle drag response [Reference Tejero, MacManus, Sanchez-Moreno and Sheaf23].

(9) \begin{align}{f_{relu}}\left( {{v_k}} \right) = {\rm{max}}\left( {0,{v_k}} \right)\end{align}

The sigmoid (Equation (10)) and hyperbolic tangent (Equation (11)) activation functions were also considered during the training process [Reference Lau and Lim37]. A well-known drawback of these functions is that the gradients at the bounds tends to 0 [Reference Haykin39], which can slow down the network training because the changes in the weights and biases of the neurons are very small.

(10) \begin{align}{f_{sigmoid}}\left( {{v_k}} \right) = \frac{1}{{1 + {e^{ - {v_k}}}}}\end{align}

(11) \begin{align}{f_{tah}}\left( {{v_k}} \right) = {\rm{tanh}}\left( {{v_k}} \right)\end{align}

The neural networks were trained using the Adam optimiser, a gradient descent algorithm known for its rapid convergence, computational efficiency, and scalability to large datasets [Reference Tejero, MacManus, Sanchez-Moreno and Sheaf23]. A full factorial for the combination of hyperparameters presented in Table 1 was considered during the training of the different ANNs. The final selection of the artificial neural network for nacelle drag prediction was based on the minimum relative root mean square error ( $\sigma $ in Equation (12)) in an independent database. The validation set was generated with the same sampling strategy as the training database, i.e. LHS for the geometric variables and anisotropic for the aerodynamic ones.

(12) \begin{align}\sigma = \sqrt {\frac{1}{N}\mathop {\mathop \sum \limits_{i = 1} }\limits^N {\epsilon ^2}} = \sqrt {\frac{1}{N}\mathop {\mathop \sum \limits_{i = 1} }\limits^N {{\left( {\frac{{C_D^{CFD} - C_D^{ANN}}}{{C_D^{CFD}}}} \right)}^2}} \end{align}

where $\epsilon $ is defined as Equation (13):

(13) \begin{align}\epsilon = \frac{{C_D^{CFD} - C_D^{ANN}}}{{C_D^{CFD}}}\end{align}

3.1 Database generation and artificial neural network training

The ANN uses eight degrees of freedom ( ${N_{DoF}}$ ) in which six are intuitive geometric variables ( ${N_{DoF - geo}}$ ), i.e. ${L_{nac}},{r_{te}},{r_{if}},{f_{{\rm{max}}}},{r_{{\rm{max}}}}$ and ${\beta _{nac}}$ (Fig. 2), and two are the aerodynamic variables of Mach number and massflow capture ratio. For the geometric degrees of freedom the Latin hypercube sampling method was used to sample the design space and approximately a seed of ${N_{s - geo}}$ = 600 nacelle shapes was generated. This provides a ratio ${N_{s - geo}}$ / ${N_{DoF - geo}}$ = 100. For each nacelle configuration, an anisotropic sampling for the two operating conditions of M and MFCR was used with a total of 102 data points per configuration. Overall, this results in the generation of a database with approximately ${N_s}$ = 60,000 CFD data samples, with a ratio of ${N_s}$ / ${N_{DoF}}$ = 7,500. Similarly, an independent dataset was generated with the same sampling techniques to quantify the ANN’s accuracy in the prediction of the nacelle drag (Equation (14)). Around ${N_{s - geo}}$ = 150 different aero-engine nacelle shapes were compiled, with a total number of approximately ${N_s}$ = 15,000 CFD data points for the validation set.

(14) \begin{align}{C_D} = \frac{{{D_{nac}}}}{{\frac{1}{2}{\rho _\infty }V_\infty ^2{A_{hi}}}}\end{align}

Different flow-field topologies are encountered throughout the design space which poses many challenges in terms of the generation of accurate low order models in aero-engine nacelle applications [Reference Heidebrecht and MacManus24]. To provide an insight of this complexity, the aerodynamic flow characteristics of two nacelle shapes compiled for the training of the neural network are showed in Fig. 4. These are presented for Mach numbers of 0.80, 0.85 and 0.90 at constant MFCR = 0.70. For Geometry 1, which has a long nacelle length with ${L_{nac}}/{r_{hi}} \approx $ 3.8, there are no shocks along the cowl at M = 0.80 (Fig. 4(a)), and has a single shock wave at M = 0.85 and 0.90 with pre-shock isentropic Mach numbers of 1.19 and 1.26, respectively (Fig. 4(c) and (e)). Conversely, the short nacelle cowl with ${L_{nac}}/{r_{hi}} \approx $ 3.0 of Geometry 2 has a well defined shock already at M = 0.80 with a pre-shock isentropic Mach number of 1.24 at the nacelle lip (Fig. 4(b)). An increase of flight velocity to M = 0.85 results in a change of the flow topology for which two shocks are created (Fig. 4(d)). The first one is located on the nacelle forebody with a pre-shock of 1.25 and the second is on the afterbody with a more benign intensity of pre-shock ${M_{is}}$ = 1.17. A further increment of M to 0.90, results in the merging of both shock topoplogies in a single strong one which induces flow separation (Fig. 4(f)).

Figure 4. Example of flow-field across the design space: Geometry 1 ( ${L_{nac}}/{r_{hi}} \approx $ 3.8) and Geometry 2 ( ${L_{nac}}/{r_{hi}} \approx $ 3.0).

The full-factorial combination of hyperparameters summarised in Table 1 was assessed to structure independent surrogate models that can approximate the nacelle drag (Equation (14)) across the design space. The best ANN model was found with 64 neurons per hidden layer, 2 hidden layers and the ReLu activation function, with a relative root mean square error of $\sigma $ = 2.9%. This is the metamodel selected in this study for preliminary nacelle design.

3.2 Validation for nacelle drag prediction

The ANN was trained with the compiled database of 60,000 samples in Python 3.7 using the Keras library and Tensorflow as backend [Reference Abadi, Barham, Chen, Chen, Davis and Dean35]. The training process was performed in an AMD EPYC 7543 32-core CPU with one NVIDIA A100 GPU. Once the model was trained, the surrogate can be interrogated with any workstation with the same Python environment. Testing has been successfully conducted on a conventional PC with 4GB of RAM (random access memory). As discussed above, the predictive accuracy of the ANN was evaluated with the independent dataset that was generated for perturbations in geometric and aerodynamic degrees of freedom. Figure 5 presents the cross-validation between the ANN predictions and the CFD results across all the design space. The overall uncertainty is $\sigma $ = 2.9%, in which 92% of the cases have an error ( $\epsilon $ in Equation (13)) below 5%, 7% of the evaluations present an error between 5% and 10%, and only 1% depict an error above 10%. The majority of the cases with $\epsilon $ $ \gt $ 10% have a MFCR $ \lt $ 0.65, which is an off-design condition. The nacelle stagnation point moves closer to the fan face and this results in a large region of acceleration along the nacelle lip that may terminate with high peak Mach numbers, strong shock waves and shock induced separation [Reference Boscagli, MacManus, Tejero, Sabnis, Babinski and Sheaf18]. For this work, an overall model uncertainty of 5% is targeted to meet the requirements of low order models for preliminary design [Reference Torenbeek40]. As such, the predictive accuracy across the design space fulfills this target ( $\sigma $ = 2.9%) and the ANN can be used at a preliminary stage of the design process.

Figure 5. ANN cross-validation of the full design space with geometric and aerodynamic degrees of freedom.

Although nacelle design is a multi-point, multi-objective optimisation problem [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31], and therefore an acceptable accuracy of the metamodel across the RSM bounds is needed (Fig. 5), the cruise condition has a notable importance in the overall design process. For this reason, the relative root mean square error ( $\sigma $ ) was also quantified for expected mid-cruise type conditions. In this respect, two extra cross-validations were performed for flight Mach numbers of 0.80 and 0.85 at fixed MFCR = 0.70 (Fig. 6). These are representative of cruise-type conditions for medium-range and long-range applications, respectively. As such, the predictive accuracy is quantified for changes in the geometric design space. For the medium-range applications with M = 0.80, the overall $\sigma $ = 2.6% in which 95.5% of the designs have a prediction error below 5% and the remaining configurations are within 5% $ \lt $ $\epsilon $ $ \lt $ 10%. In addition, the relative root mean square error for long-range applications with M = 0.85 and MFCR = 0.70 is $\sigma $ = 3.2% in which 89.2% have $\epsilon $ $ \lt $ 5%, 10.4% of the nacelle are within 5% $ \lt $ $\epsilon $ $ \lt $ 10% and 0.4% have $\epsilon $ $ \gt $ 10%. As it could be envisaged, the relative root mean square error ( $\sigma $ ) is larger for the high Mach number due to the increased non-linearity of this condition.

Figure 6. ANN cross validation for medium- and long-range applications at mid-cruise conditions.

To show how the nacelle drag changes across the design space, Fig. 7 presents the drag variation as a function of the aerodynamic degrees of freedom (M and MFCR) for three different nacelle shapes from the validation dataset. They have ${L_{nac}}/{r_{hi}} \approx $ 3.8, 3.0 and 2.2, and are referred to in Fig. 7 as design A, B and C, respectively. These configurations cover a wide range of the design space that is representative of conventional and compact aero-engine nacelles for medium- and long-range applications. For the configuration with ${L_{nac}}/{r_{hi}} \approx $ 3.8 (design A) the changes in nacelle drag as a function of Mach number and massflow capture ratio are benign. For a fixed MFCR, ${C_D}$ smoothly increases as a function of flight Mach number caused by compressibility effects and shock-waves that are formed on the nacelle (Figs. 7(a) and (d)). Similarly, at a constant M the nacelle drag coefficient increases when MFCR is reduced due to the larger flow acceleration around the nacelle lip caused by the movement of the stagnation point towards the fan. For this nacelle configuration the ANN successfully predicts the drag changes across the wide aerodynamic design space within 0.55 $ \lt $ M $ \lt $ 0.925 and 0.5 $ \lt $ MFCR $ \lt $ 0.93 with a maximum absolute error of $\epsilon $ = 4.2% (Fig. 7(g)). For the nacelle design with ${L_{nac}}/{r_{hi}} \approx $ 3.0 (configuration B) the influence on ${C_D}$ of M and MFCR is different relative to A. For the design B the sensitivity to a change of MFCR is modest and the changes on ${C_D}$ are mainly dominated by Mach number (Fig. 7(b) and (e)). For a MFCR below 0.65, the drag rise curve has a local minima caused by the changes on the shock topology. For benign values of M, the nacelle aerodynamics are governed by a single shock. An increment in Mach number results in the change of the flow-field topology to a double shock structure with relatively low values of pre-shock ${M_{is}}$ . This effect reduces the nacelle drag due to a reduction in the wave drag. Increasing further the freestream Mach number results in the combination of both shocks into a single strong one that has associated high levels of wave drag [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31] and a concomitant larger nacelle drag. Modelling this non-linearity with a generalised low order model that contains configurations with such characteristics (e.g. design B) as well as designs with benign changes on ${C_D}$ (e.g. design A) is complex. For this reason, previous Kriging methods filtered and smoothed non-monotonic drag-rise behaviours to enable the generation of a RSM with an acceptable predictive accuracy [Reference Heidebrecht and MacManus24]. However, the proposed ANN is able to include the described nacelle drag characteristics without simplifications in the drag response of the nacelle. For the design B, the maximum absolute predictive error across the aerodynamic DoF is $\epsilon $ = 10% at M $ \approx $ 0.86 and MFCR $ \approx $ 0.6 (Fig. 7(h)). Lastly, the configuration C with ${L_{nac}}/{r_{hi}} \approx $ 2.2 has a notable sensitivity to flight Mach number (Figs. 7(c) and (f)). This is caused by the high curvature distribution along the fancowl of this short nacelle which induces a strong flow acceleration. For this reason, the drag sensitivity to Mach number is greater than for designs A and B. For example, for flight Mach numbers above 0.85, the design C is operating post drag rise across the different MFCR which poses many challenges for a RSM due to the associated large drag gradients. Despite its complexity, the ANN predicts the changes on ${C_D}$ as a function of M and MFCR with a maximum absolute error of 4.7% (Fig. 7(i)).

Figure 7. Nacelle drag as a function of Mach number and massflow capture ratio for 3 nacelle samples: Design A ( ${L_{nac}}/{r_{hi}} \approx $ 3.8), Design B ( ${L_{nac}}/{r_{hi}} \approx $ 3.0) and Design C ( ${L_{nac}}/{r_{hi}} \approx $ 2.2).

Within the context of preliminary nacelle design, it is important to note that the normalised nacelle length ( ${L_{nac}}/{r_{hi}}$ ) is a key geometric variable because it has the biggest influence on surface area and, thus, nacelle drag and mass. Although the target may be to shorten the nacelle as much as possible, the lower bound is usually determined by the intake and exhaust length requirements [Reference Goulos, Otter, Stankowski, Macmanus, Grech and Sheaf41, Reference Hueso-Rebasa, MacManus, Tejero, Goulos, Sanchez-Moreno and Sheaf42]. Trade-off studies on this variable are needed to ensure that low values of ${L_{nac}}/{r_{hi}}$ do not result in nacelle drag penalties caused by shock losses that arise due to fancowls with high curvature and large flow accelerations [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31]. In addition, the non-dimensional nacelle trailing edge ( ${r_{te}}/{r_{hi}}$ ) is constrained by the exhaust shape and its flow requirements. The lower bound of the nacelle boat-tail angle ( ${\beta _{nac}}$ ) is imposed by the outer bypass duct requirements and the normalised maximum radius ( ${r_{{\rm{max}}}}/{r_{hi}}$ ) is usually bounded by the compromise between installed ground clearance considerations and the requirement of keep-out-zones for auxiliary components outside of the fan case, e.g. thrust reverser unit. For the purpose of providing an insight of how the neural network may be deployed in the preliminary design phase, the normalised geometric variables ${L_{nac}}/{r_{hi}}$ , ${r_{te}}/{r_{hi}}$ , ${\beta _{nac}}$ and ${r_{{\rm{max}}}}/{r_{hi}}$ are fixed, and the two remaining ones, i.e. ${f_{{\rm{max}}}}$ and ${r_{if}}{/_{hi}}$ , vary to quantify their influence on the nacelle drag characteristics. Figure 8 presents the effect of ${f_{{\rm{max}}}}$ and ${r_{if}}/{r_{hi}}$ on ${C_D}$ for two different configurations with lengths of ${L_{nac}}/{r_{hi}} \approx $ 3.8 (Figs. 8(a) and (c)) and ${L_{nac}}/{r_{hi}} \approx $ 3.0 (Figs. 8(b) and (d)). This is shown for representative mid-cruise conditions of long-range applications with M = 0.85 and MFCR = 0.70. The maps with the ANN are compared with higher-fidelity CFD data, where it is highlighted the good agreement in nacelle drag between the numerical simulations and the ANN predictions. Across the range of ${f_{{\rm{max}}}}$ - ${r_{if}}/{r_{hi}}$ , there is a maximum error of 2.2% and 3.0% for the configuration with ${L_{nac}}/{r_{hi}} \approx $ 3.8 and ${L_{nac}}/{r_{hi}} \approx $ 3.0, respectively. This deviation is within an overall $\sigma $ = 3.2% of the model for mid-cruise conditions (Fig. 6(b)). Across the ${f_{{\rm{max}}}}$ - ${r_{if}}/{r_{hi}}$ space, the nacelle drag changes by approximately 96% for the short design ( ${L_{nac}}/{r_{hi}} \approx $ 3.0) and by 30% for the long configuration ( ${L_{nac}}/{r_{hi}} \approx $ 3.8). This greater sensitivity of the compact nacelle is expected to be due to the higher non-linearity of this region of the design space [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31]. The location of the minimum ${C_D}$ as a function of ${f_{{\rm{max}}}}$ and ${r_{if}}/{r_{hi}}$ changes with the nacelle style (Fig. 8) and is well predicted by the ANN, which establishes confidence in the low order model for the design of aero-engine nacelles.

Figure 8. Nacelle drag as a function of the intuitive design variables ${f_{{\rm{max}}}}$ and ${r_{if}}$ for two nacelle samples with ${L_{nac}}/{r_{hi}} \approx $ 3.8 and 3.0.

3.3 Validation for optimisation studies

Although it is important to ensure that a low order model has a sufficient predictive accuracy (Section 3.2), another key aspect is that the gradients of the performance metrics across the design space are well predicted. This is to ensure that key decisions in terms of the feasibility of new concepts are correct. Furthermore, it is fundamental that the geometric design variables of optimal configurations are appropriately identified to ensure confidence in their imposition as constraints in the design and optimisation of other sub-systems. For this reason, it is required to test the capability of response surface models in optimisation environments. In this respect, a set of surrogate-based optimisations were carried out with the neural network model and compared with the equivalent computationally expensive CFD-based ones [Reference Tejero, Christie, MacManus and Sheaf17, Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31]. As previously described, two of the nacelle design variables ( ${L_{nac}}/{r_{hi}}$ and ${r_{te}}/{r_{hi}}$ ) are usually fixed by the intake and exhaust housing components. For this reason a range of independent multi-point, multi-objective optimisations within the range of 2.7 $ \lt $ ${L_{nac}}/{r_{hi}}$ $ \lt $ 3.6 and 0.90 $ \lt $ ${r_{te}}/{r_{hi}}$ $ \lt $ 1.0 were performed. During this process, the other variables, i.e. ${r_{if}}/{r_{hi}},{f_{{\rm{max}}}},{r_{{\rm{max}}}}/{r_{hi}}$ and ${\beta _{nac}}$ (Fig. 2), varied. The optimisations were performed with a well-established approach [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31], in which three different conditions that are encountered in the cruise segment were evaluated. They are mid-cruise as well as the sensitivity to Mach number and mass flow capture ratio (Table 2). This analysis was carried out for long-range applications because it is the most challenging scenario for a low order model in civil aero-engine nacelle applications.

The optimisation process for both ANN-based and CFD-based approaches is coupled with the OMOPSO algorithm [Reference Sierra and Coello Coello43]. The selection of this gradient-free method is based on its suitability for aero-engine nacelle applications [Reference Sanchez-Moreno, MacManus, Tejero, Matesanz-Garcia and Sheaf44]. The process starts with a design space exploration (DSE) based on a LHS sampling to efficiently cover the design space. It is formed by 400 different designs to provide a ratio between the DSE size and degrees of freedom of 100. Subsequent generations of the OMOPSO process are followed by 50 nacelle designs. The optimisation convergence is monitored by means of the hypervolume indicator. The optimisation is stopped when the hypervolume changes below 1.0% in three consecutive generations.

Table 2. Flight conditions considered during the multi-point, multi-objective nacelle optimisation process

To demonstrate the capability of the nacelle design approach, CFD-based and ANN-based optimisations are initially performed for a compact aero-engine with ${L_{nac}}/{r_{hi}} = 3.0$ and ${r_{te}}/{r_{hi}}$ =1.0. For the same configuration, a previous RSM-based optimisation with a Gaussian regression method based on Kriging highlighted the challenges of identifying the same optimal design space as a computationally expensive CFD in-the-loop approach [Reference Tejero, MacManus and Sheaf21]. This was caused by the non-linearity of the nacelle drag characteristics of this compact architecture and the difficulties associated with Kriging modelling. After successful convergence of the multi-point, multi-objective optimisation with the ANN surrogate-based and the CFD-based strategies, a set of Pareto optimal solutions were identified (Fig. 9(a)). The 3D surface is projected in the ${C_{D - cruise}}$ - ${C_{D - iM}}$ space, where the trade-off between the two objective function can be observed. The comparison of the Pareto surface for both optimisation strategies reveals the capability of the ANN-based to identify a similar shape. The hypervolume of the computationally expensive CFD in-the-loop is only 2.6% larger than the RSM-based one. Relative to the CFD in-the-loop, the minimum drag ${C_{D - cruise}}$ identified by the ANN is 0.2% greater. This changed to 1.0% for the minimum achievable ${C_{D - iM}}$ and a reduction of 0.2% for ${C_{D - EoC}}$ (Fig. 9(b)). These differences are within the predictive uncertainty $\sigma $ = 2.9% of the ANN model (Fig. (5)). The greatest discrepancy is identified for the increased Mach number case (Table 2) with M = 0.87 due to the expected greater uncertainty of the ANN as the freestream Mach number increases. Another key aspect of a surrogate-based multi-point, multi-objective optimisation is to ensure that it converges to the same regions of the design space. For all the design variables ( ${f_{{\rm{max}}}},{r_{{\rm{max}}}}/{r_{hi}}$ , ${r_{if}}/{r_{hi}}$ and ${\beta _{nac}}$ ), the normalised mean values of the Pareto front are similar between the CFD-based and RSM-based multi-point, multi-objectives optimisations (Fig. 9(c)). The maximum normalised difference is of 0.04 for ${f_{{\rm{max}}}}$ , which reduces to 0.001, 0.02 and 0.008 for ${r_{{\rm{max}}}}/{r_{hi}}$ , ${r_{if}}/{r_{hi}}$ and ${\beta _{nac}}$ , respectively. As such, it can be concluded that the neural network is able to drive the optimisation process to similar regions of the design space relative to the computationally expensive CFD in-the-loop approach. This establishes confidence for its deployment for optimisation purposes. It is important to note that previous studies based on CFD-based optimisations with gradient-free methods concluded that the convergence of the multi-point, multi-objective optimisation required the evaluation of approximately 1,850 designs. These were composed by 400 nacelles in the design space exploration (DSE) followed by 29 generations of 50 individuals each [Reference Robinson, MacManus, Heidebrecht and Grech45]. The same settings were used in this work for the CFD-based optimisation, which results in a total of 5,550 independent CFD evaluations, i.e. 1,850 designs with three flight conditions per design. This is usually not acceptable within a preliminary design stage due to the large overhead in terms of computing requirements and overall time to perform the optimisation.

Table 3. Multi-point, multi-objective optimisation for aero-engine nacelles

Figure 9. Comparison for the ANN- and CFD-based multi-point, multi-objective optimisations.

Having established confidence that the developed approach identifies similar designs as the computationally expensive CFD-based approach, the method was used for a wide range of nacelle configurations with different ${L_{nac}}/{r_{hi}}$ and ${r_{te}}/{r_{hi}}$ . In this respect, a full factorial combination for values of ${L_{nac}}/{r_{hi}}$ = 2.7, 3.0, 3.3 and 3.6, and ${r_{te}}/{r_{hi}}$ = 0.90, 0.95, 1.0 were considered. This is to quantify if the surrogate-based approach can predict the nacelle drag gradients across the design space as well as to identify the associated geometric design variables of the optimal configurations. For each of the 12 nacelle configurations, an independent multi-point, multi-objective optimisation (Table 2) was carried out in which the process was driven with ANN predictions and CFD simulations. For all the optimisations, the process resulted in a set of Pareto optimal solutions upon which a nacelle was downselected (Table 3). The downselection criteria can be modified according to the parameters ${K_1}$ and ${K_2}$ (Table 3) depending on the requirements and relative importance that the designer wants to provide to the sensitivity to Mach number ( ${C_{D - iM}}$ ) and MFCR ( ${C_{D - EoC}}$ ). The overall process results in the generation of drag maps to quantify the nacelle drag changes across the design space for different ${L_{nac}}/{r_{hi}}$ and ${r_{te}}/{r_{hi}}$ . For the CFD in-the-loop optimisation, the nacelle mid-cruise drag changes by about 16% across the design space (Fig. 10). The non-dimensional nacelle length ( ${L_{nac}}/{r_{hi}}$ ) has a larger impact on the drag variations than ${r_{te}}/{r_{hi}}$ . For a fixed ${r_{te}}/{r_{hi}}$ = 1.0, ${C_{D - cruise}}$ reduces by 13.4% when the nacelle length is shortened from ${L_{nac}}/{r_{hi}}$ = 3.6 to 3.1. For a conventional configuration with ${L_{nac}}/{r_{hi}}$ = 3.6, the changes in ${r_{te}}/{r_{hi}}$ between 1.0 and 0.90 have a negligible effect on the mid-cruise nacelle drag. Conversely, for a compact aero-engine nacelle with ${L_{nac}}/{r_{hi}}$ = 3.0, the nacelle drag increases by 4.0% when the trailing edge radius is moved inboards from ${r_{te}}/{r_{hi}}$ = 1.0 to 0.90. For the short nacelle configuration with ${L_{nac}}/{r_{hi}}$ = 2.8, there was not a design that fulfilled the downselection criteria (Table 3) for the most compact ${r_{te}}/{r_{hi}}$ = 0.90. Similar mid-cruise drag changes were identified for the ANN-based optimisations with a variation of about 15% across the range of ${L_{nac}}/{r_{hi}}$ and ${r_{te}}/{r_{hi}}$ investigated (Fig. 10). The effect of ${L_{nac}}/{r_{hi}}$ on nacelle drag is also comparable to the findings of the CFD in-the-loop optimisations. For the RSM-based cases, the mid-cruise nacelle drag decreased by 13% when the nacelle length reduced from ${L_{nac}}/{r_{hi}}$ = 3.6 to 3.1 at fixed ${r_{te}}/{r_{hi}}$ = 1.0. This is close to the findings from the computationally expensive CFD studies that showcased a reduction of 13.4%. Lastly, the relative importance of the effect of the nacelle trailing edge ( ${r_{te}}/{r_{hi}}$ ) at constant nacelle length was also similar between the ANN and CFD optimisation approaches (Fig. 10). It is demonstrated that the low order model predicts similar changes on nacelle drag across a wide dimensional space, which builds confidence in its deployment within a preliminary nacelle design process.

Figure 10. Mid-cruise nacelle drag changes as a function of ${L_{nac}}/{r_{hi}}$ and ${r_{te}}/{r_{hi}}$ for downselected designs from the multi-point, multi-objective optimisations.

Other key consideration of surrogate-based multi-point, multi-objective optimisation is that the process converges to the same parts of the design space as a computationally expensive CFD in-the-loop method. In this respect, the geometric degrees of freedom of the previous downselected designs are compared for both approaches. Figures 11 and 12 show the normalised values of ${f_{{\rm{max}}}}$ and ${r_{{\rm{max}}}}/{r_{hi}}$ , which have a larger impact on the nacelle drag characteristics than ${r_{if}}/{r_{hi}}$ and ${\beta _{nac}}$ [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31]. This representation is useful to quantify the changes of the intuitive nacelle design variables to derive nacelle design guidelines and design styles at a preliminary stage of the design process. The maps can also be used as constraints for the design and optimisation of other sub-systems. The design variables ${f_{{\rm{max}}}}$ and ${r_{{\rm{max}}}}/{r_{hi}}$ increase as the nacelle is shortened and the trailing edge radius is moved outboards. This is to reduce the nacelle curvature and the flow acceleration which has a direct impact on the overall nacelle wave drag [Reference Swarthout, MacManus, Tejero, Matesanz-Garcia, Boscagli and Sheaf31]. The absolute value, as well as the gradients, across the design space for both intuitive variables are very similar between the ANN- and CFD-based optimisations. It demonstrates that the developed artificial neural network can be deployed with confidence for multi-point, multi-objective optimisation studies of aero-engines nacelles.

Figure 11. Changes of the normalised intuitive variable ${f_{{\rm{max}}}}$ as a function of ${L_{nac}}/{r_{hi}}$ and ${r_{te}}/{r_{hi}}$ for downselected designs from the multi-point, multi-objective optimisations.

Figure 12. Changes of the normalised intuitive variable ${r_{{\rm{max}}}}/{r_{hi}}$ as a function of ${L_{nac}}/{r_{hi}}$ and ${r_{te}}/{r_{hi}}$ for downselected designs from the multi-point, multi-objective optimisations.