Artificial neural network-based streamline tracing strategy applied to hypersonic waverider design

Contemporary mission requirements of long-range maneuverable hypersonic vehicles for space, military, and transport applications demand innovations in vehicle design. Unique aerothermodynamic challenges in the hypersonic flow regime necessitate a fully integrated hypersonic vehicle geometry.^1,2 The hypersonic waverider is a special aerodynamic shape whose leading edges are in close proximity to the shock wave produced by the body (as if the body is riding the shock wave). The lower surface of the vehicle encapsulates a region of high pressure, providing a significantly high lift to drag ratio essential for modern-day applications. Under the integrated vehicle design philosophy, large portions of the vehicle forebody are contoured as an intake to efficiently deliver compressed air to the air-breathing propulsion system. Streamline tracing forms an indispensable technique in designing high-performance intakes and waveriders.

Hypersonic intakes derived from the axisymmetric converging flow field, known as the Busemann flow field, have been shown to give high-efficiency compression.^3,4 In a detailed chapter, Molder⁵ discussed the benefits and procedures to calculate the Busemann intake. Streamline tracing is an essential step in the procedure. Furthermore, the streamline tracing technique can be exploited to obtain intake shapes that suitably converge from the entry to an exit of the shape desired at the combustor resulting in different arrangements, such as the sugar scoop-shaped intake. The startability of the sugar scoop-shaped intake and its control by means of a sliding door has been experimentally tested at the Virginia Supersonic Windtunnel at Mach number 4.⁶ 

Küchemann⁷ in an overarching study emphasized the peculiar challenges of designing hypersonic vehicles in comparison to subsonic or supersonic vehicles and, in this context, highlighting the need for lifting bodies, such as the waverider, and detailed several waverider configurations. The waverider concept was first introduced by Nonweiler⁸ with the caret shaped waverider. Typically, the waverider consists of a freestream surface parallel to the freestream and a compression surface that forms the underbelly of the vehicle. The leading edge, which is the intersection of the freestream surface and the compression surface, rides the shock wave of the body, capturing a high pressure region under the compression surface, thereby generating a significantly higher lift. The base surface comprises the freestream surface (base) curve and the compression surface curve in a plane perpendicular to the freestream. Waverider shapes are generated from a basic flow field by projecting the base curve onto the shock and using streamline tracing to generate the compression surface, as depicted pictorially in Fig. 1. The wedge based flow field generates an oblique shock, which is easily computed. The caret shaped waverider and the power-law waverider are typical examples of wedge-based waveriders.⁹ Besides the lift to drag ratio, L/D, the volume contained within the waverider is important from the payload perspective. Hence, good volumetric efficiency is essential. Waveriders can be generated from conical flow fields, which can be computed using the Taylor–Maccoll equations. In some cases, generic three-dimensional shapes have also been used as the basic flow field. Recently, state of the art in waverider design methodologies have been comprehensively described in a review article.¹⁰ 

Wedge derived waveriders formed the basis of developing the conceptual design of a complete air-breathing propulsion hypersonic vehicle by Ferguson et al.¹¹ A modular design where multiple waverider configurations were combined to generate a star-shaped aerodynamic surface was described. However, requirements for larger volumetric efficiencies led to a shift toward cone based waverider designs in a later study.¹² Jones et al.¹³ described the generation of simple cone based waveriders from an axisymmetric right circular cone flow field and classified two kinds of waveriders, one where the apex of the waverider coincides with the axis of the cone and the second where it is offset from the cone axis. A high degree of design flexibility is required to meet the several competing objectives of hypersonic vehicles, for example, the need for bluntness at the leading edges to avoid severe aerodynamic heating. Therefore, many improvements to the design process continue to be made even today. A generalized inverse design method, where the shock wave shape is given and axisymmetric flow fields are utilized to generate the given shock curve along osculating planes, significantly enhances the design space for waveriders.¹⁴ The osculating cone method is a particular case of the general method, which has been used extensively in recent times. In further development, the requirement of the same axisymmetric shape for the flow field at each osculating plane has been relaxed, leading to the osculating flow field method for waverider design.¹⁵ Ding et al.^16,17 proposed a simplification to the streamline tracing method of the conical flow field, and the resulting waverider has shown an increase in volumetric efficiency. However, flow spillage has been observed at the leading edge. The method of computing the axisymmetric flow field plays a role in determining the rapidity and accuracy of waverider design and parametric analysis. Analytical solutions from hypersonic small disturbance theory enables rapid assessment of a large design space, but their applicability is restricted to small cone angles.^18,19 In the majority of the studies, Taylor–Maccoll relations are solved to obtain the axisymmetric flow field. High fidelity Euler solutions allow the generation of waveriders from a more general class of conical shapes, which need not be restricted to right circular cones, but they come with high computing costs and time.²⁰ Chen et al.²¹ improvised the osculating cone technique by adjusting the radius of the cone to achieve a target volume of the waverider. Attempts have been made to make waverider design suitable to a wide range of Mach numbers by providing a Mach number profile at each osculating plane along the prescribed shock curve²² or by discretizing the leading edge curve for different Mach numbers.²³ The streamline tracing technique has been extended to waverider generation from a wide variety of basic flows, including wedge–cone combination,²⁴ power-law blunt bodies,¹⁹ and von Karman ogive,²⁵ to name a few. In some instances, viscous effects have been considered by using integral boundary layer equations after solving the inviscid flow field over the basic shape before finally deriving the waverider shape.^19,26,27 Integration of the intake with the waverider remains a key challenge. Novel strategies have been developed: in one study, a combined axisymmetric base flow consisting of a cone with the intake cowl has been taken as the basic flow field,²⁸ and in another, a suitable merging of inward turning and outward turning axisymmetric flow fields has been achieved.²⁹ Thus, with multiple competing objectives to be satisfied, the waverider design principles continue to evolve. Generally, the newly developed waverider design method is compared with existing methods using inviscid and Reynolds Averaged Navier–Stokes (RANS) computational fluid dynamics (CFD) computations.^16,21,25 Rapid computation of waverider geometry is crucial to explore the design space for optimization purposes, which requires efficient parameterization and predictive capabilities.

Modern-day data-driven Artificial Intelligence (AI) tools have revolutionized data analysis and prediction capabilities. MATLAB provides several algorithms for Machine Learning (ML), Artificial Neural Networks (ANN), and AI, which are elaborately described in Ref. 30. The ANN consists of interconnected layers of nodes that function analogous to biological neurons. They are known to excellently approximate functions between multiple inputs to multiple outputs of heterogeneous categories. The ANN learns from a known database presented to it by calculating the weights of the interconnections between nodes of different layers by minimizing a suitably defined error function. Post learning, the ANN can be used as a predictive tool. The general architecture of an ANN consists of one input layer, one output layer, and several hidden layers, as depicted in Fig. 2. The overall workflow for training the ANN is also represented in the figure. The ANN has been deployed in the solution of differential equations³¹ and the numerical thermochemical computations for hypersonic flows.³² Kutz³³ briefly described the rapid rise in the use of Deep Neural Networks (DNNs) for turbulence modeling and prediction. Miyanwala and Jaiman successfully used DNN to solve the Navier–Stokes equation for the unsteady vortex shedding problem behind a cylinder.³⁴ Aerodynamic shape optimization requires large-scale computations of several parameters, which is computationally expensive and time-consuming. Flow field reconstruction from experimental and high fidelity CFD data has also gained significantly from deep learning techniques, for example, in the prediction of flow field over airfoils³⁵ and the velocity field in scramjet isolator.³⁶ A generalized ANN-based model of the supersonic ejector has been developed by the author’s group, which has the capability to predict performance for different working fluids.³⁷ 

The hypersonic vehicle design and optimization process are essentially multi-disciplinary, involving strong interactions, such as fluid–structure, thermal–structure, fluid–thermal–structure-control, and eventually with the overall trajectory, as detailed in the Ref. 38. Aerothermodynamic interactions in the rarefied hypersonic domain especially at high altitudes need high fidelity numerical codes, such as the Direct Simulation Monte Carlo (DSMC) approach,³⁹ which in a later work was utilized to optimize the geometry by using the Bezier curve as a tool for parameterization.⁴⁰ An increase in computing power has led to the use of high fidelity numerical computations to predict vehicle performance. A block diagram indicating an overall optimization process of the waverider, where the waverider geometry generation is dovetailed to a high fidelity numerical analysis that considers multiphysics interactions, is shown in Fig. 3. The waverider geometry generation can include CFD simulations, which makes it computationally expensive. Objective functions evaluated at the end of the high fidelity numerical computation enable a feedback that modifies the input parameters to the waverider geometry generation via an optimizer. This iterative process is highly computationally expensive. Often, a surrogate model is trained using a discrete sample space of high fidelity numerical computations, as illustrated in Fig. 4. Zhang et al.⁴¹ employed the response surface analysis technique as a surrogate model in optimizing a generic hypersonic vehicle. Similarly, the ANN/CNN can also be employed as a surrogate model. ANN has been used as a surrogate model by learning from a limited dataset to predict a large design space as in the cases of the airfoil and wing geometry.^42–44 Fujio and Ogawa⁴⁵ utilized a surrogate assisted evolutionary optimization framework in optimizing a streamline traced sugar-scoop type intake. Once a surrogate model is employed to generate the data for objective functions, there arises an immediate need to speed up the geometry generation process as well. Therefore, an ANN derived waverider geometry generation process can bring about further increase in computational speed, as depicted in Fig. 5. Hence, the objective of this work is to develop a novel ANN derived waverider geometry generation technique. The ANN based streamline tracing strategy developed in this work can be incorporated in increasingly expensive simulations, such as the RANS based CFD on basic flow fields, and thereby achieve significant reduction in geometry generation times. However, for the purpose of development and comparison with standard approaches, we utilize the well-known Taylor–Maccoll equations for axisymmetric flows in this work.

Hypersonic waveriders are the preferred shapes for long-range manoeuvrable hypersonic vehicles. Predominantly axisymmetric cone flow fields solved by the Taylor–Maccoll equations are utilized to design waveriders. For a single waverider shape, several streamline tracing calculations have to be carried out to define the compression surface, which involves marching solutions to the streamline differential equations. When considering the optimization of the waverider for multiple objectives, such as high aerodynamic efficiency, volumetric efficiency, and low aerodynamic heating, effective parameterization and computation of the waverider geometry over a large design space must be accomplished in the shortest duration. However, solving differential equations is computationally expensive and time-consuming. ANN has been effectively used as an approximation to function in the solution of differential equations. However, the application of ANN for streamline tracing leading to rapid waverider shape generation has not been explored in the literature, which motivated the current investigation. First, a parameterized form of the streamline equations in terms of a polynomial fit with high prediction accuracy is obtained from the solution of Taylor–Maccoll equations for a wide range of Mach numbers. The coefficients of the polynomial fit are then predicted using an ANN, which has been trained using the extensive database generated. Any change in the waverider generation method affects the flow field to a certain extent. For example, the simplified method, which uses straight streamlines parallel to the cone surface, produces waveriders that can develop tip spillage. Hence, RANS based CFD is performed on waveriders to verify the flow field obtained from the new geometry generation process. Waveriders generated from the standard cone method, simplified cone method, and the proposed ANN method are compared for shapes and final flow fields by CFD solutions of RANS equations. We have successfully achieved highly accurate waverider geometries using the ANN method with significant reduction in computational time.

The inviscid axisymmetric conical flow field over a right circular cone in supersonic flow is first solved using the Taylor–Maccoll (TM) equation given in Eq. (1), where V_r and V_θ are the velocity components in the radial and azimuthal directions. Equation (2) gives the relation for V_θ in terms of V_r, where V′ is defined by Eq. (3). The boundary conditions are the oblique shock jump condition at the shock and flow tangency condition at the wall. In this work, the TM equation is solved using MATLAB with a fourth-order Runge–Kutta function to obtain the flow field between the shock and the cone wall, as illustrated in Fig. 6,

Following the solution of the TM equation, the streamline differential equation represented in Eq. (4) can be evaluated by numerically marching from points located on the shock to obtain the streamline points. A schematic representation of the streamlines traced from the right circular cone shock is depicted in Fig. 6. The embedded flowchart represents the steps to obtain the streamlines from inputs (M, β) to the streamline points,

The base curve has to be specified to obtain the waverider geometry using the cone derived waverider method. The base curve is projected onto the shock by generators parallel to the freestream to obtain the leading edge. Thereafter, streamline tracing is carried out from leading edge locations on the shock back to the base plane to generate the compression surface. Two different streamline tracing techniques have been followed: in the standard cone derived waverider technique, the streamline equation [Eq. (4)] is solved, whereas in the simplified technique, streamlines are taken tangentially to the base cone surface to easily obtain the compression surface points. Algorithm 1 details the steps for both the standard and simplified methods of cone derived waverider geometry generation. Figure 7 pictorially represents the differences in the streamlines generated using the standard technique and the simplified technique.

The base curve equation employed for waverider design is presented about the symmetry plane in Fig. 8. l_u forms the length of the flat portion of the curve, while l is the width of the waverider. The curved portion of the curve is given by the power law equation. h is the offset of the curve from the cone axis and m is the power. The value of b can be determined for given l_u, l, and h.

The standard design procedure described in Sec. II A requires the solution of TM and streamline differential equations, which are computationally expensive. A novel ANN based strategy is introduced in this work to trace the streamlines and subsequently generate the waverider geometry. Figure 9 compares the standard procedure represented as a flowchart with the novel ANN based technique. Two blocks, solving the TM equation and streamline equation, in the conventional technique are replaced by a single block involving utilization of a trained ANN to predict the streamlines directly from the given inputs—Mach number (M), shock angle (β), and the location along the shock (x_s/L).

The generation of the trained ANN involves parameterization of the streamline curve using a polynomial fit followed by training the ANN for the coefficients of the polynomial for given inputs of Mach number (M), shock angle (β), and streamline starting location on the shock (x_s/L),

First, a large dataset of parameterized streamline curve coefficients is generated by solving the TM equation and the streamline equation. Equation (5) represents the parameterized form of the streamline curve approximated as a polynomial where X = x/L, Y = y/L, and (x, y) are points on the streamline and L is the length of the cone. The coefficients of the polynomial curve fit, C_n = f(M, β, x_s/L), are unique functions of only M, β, and X_s. Y_s is not independent since Y_s = X_s tan(β).

For a given combination of (M, β, X_s), the TM equation and the streamline equations are solved to obtain the points on the streamline curve. A polynomial regression fit is carried out to obtain the polynomial expression for the streamline with the corresponding coefficients (C₀, C₁, C₂, …, C_n). The process is repeated for discrete but a wide range of (M, β, X_s) combinations within limits of the valid conical shock solutions. This results in the dataset that is further used for training the ANN. The process is represented as a flowchart in Fig. 10.

Different orders of the polynomial are tested for accuracy and compactness of the resulting dataset. The third-order polynomial is found to be the best in both measures. More details on the variation of errors with degree of polynomial are described in Sec. IV. Moreover, the variation in fitness of the polynomial due to changes in M and β is found to be minimal.

A fully connected feed-forward artificial neural network with the tanh activation function is trained for (M, β, X_s) as inputs and the coefficients of the polynomial fit (C₀, C₁, C₂, …, C_n) as outputs. The tanh function is used as the activation function at each node in the network. The entire dataset is divided into training, testing, and validation sets. The training and testing datasets are used to train the ANN and evaluate its accuracy by means of back-propagation and adjustment of weights and biases at each node for error minimization. The validation set is finally used to obtain the global performance in terms of accuracy of prediction after obtaining the ANN model. Ultimately, the validated ANN model is utilized in the waverider design process. The steps followed in obtaining the ANN streamline tracing model are outlined in the flowchart represented in Fig. 11.

A study is carried out by varying the number of hidden layers and the number of neurons in each hidden layer to obtain a compact network architecture with high accuracy. The influence of choosing one among three different optimization algorithms, namely, Levenberg Marquardt, Bayesian regularization, and scaled conjugate gradient are also evaluated. The finalized ANN model consists of a single hidden layer network with 32 neurons and Bayesian regularization optimizer.

The novel ANN streamline tracing model is then integrated with the cone derived waverider approach, where the streamline curve equation arising from specific points on the leading edge is directly predicted by the ANN model. The point cloud representing the waverider surface is generated from the known curves of the freestream surface and the compression surface. The waverider geometry obtained from the point cloud is taken up for flow analysis using ANSYS CFD tools.

A comparative study is carried out on three configurations of cone derived waveriders modeled by the standard, the simplified, and the novel ANN based methodology. The models are designed for a base cone angle of 12° and base curve, as presented in Sec. II A 4. Three views of the standard waverider model are presented in Fig. 12. The shapes of the waverider generated from the simplified approach and the ANN based approach are similar except for differences in the coordinates, which is described in Sec. IV C. Furthermore, the models are meshed in the software ICEM CFD. Due to the symmetric nature of the geometry, one-half of the model is considered for simulation. Figures 13(a) and 13(b) pictorially represent the structured grid on the base plane and symmetry plane of the waverider. A y+ of 30 is ensured for the near-wall mesh in tune with the requirements of the Spalart–Allmaras turbulence model used.

The freestream condition is set to Mach 5.5 for an altitude of 20.8 km in this study. The atmospheric pressure and temperature at said altitude are 4786.5 Pa and 217.65 K. The reference length and area values are the length of the waverider and its planform area, respectively. Reynold’s number estimated for the reference length of 0.2546 m is 1.11 × 10⁶. The density-based implicitly coupled Reynolds Averaged Navier–Stokes (RANS) solver is used to numerically simulate steady-state high-speed flow in the commercial software, ANSYS Fluent.

Spalart–Allmaras (SA) single equation eddy-viscosity turbulence model developed by Spalart and Allmaras⁴⁶ is widely used in aerospace applications. Validity of the SA model for high-speed flows is examined by Paciorri et al.⁴⁷ for hollow cylinder-flare and hyperboloid flare configurations. The pressure measurements of the two models at Mach 5 and Mach 6.8, respectively, were found to be in good agreement with experimental data. Roy and Blottner⁴⁸ in an exhaustive survey consolidates the results of various turbulence models used for axisymmetric models in high-speed flows. SA performed well in flow fields involving attached flows and wakes.

The Spalart–Allmaras turbulence model with default coefficient settings as given by Matsson⁴⁹ is utilized to calculate the viscous flow field solution. The Roe-Flux-Difference Splitting (FDS) method, which is the second order spatially accurate upwind scheme, is applied to calculate the fluxes, and the least squares cell method is used in the gradient calculation. Freestream air is considered to be calorically perfect with the viscosity coefficient evaluated by Sutherland’s law using the three-coefficient method. Convergence of the solution is assumed to be achieved when the residuals (rms) of continuity, energy, the three velocities fall below 10⁻⁴. Furthermore, integral forces over the body were monitored, and it was ensured that at convergence, the force computation also had stabilized.

The boundary conditions of the flow field are schematically presented in Fig. 14. While the far field is conditioned to freestream flow, a no-slip wall boundary condition is imposed on the model surface. The base plane is a non-reflecting pressure outlet boundary.

Grid independence study was carried out for all three geometries. Results for the standard waverider geometry are presented here. Three structured grids are compared to check grid independence. The variation of coefficient of pressure (C_p) along the length of the lower compression surface of the waverider is shown in Fig. 15. A zoomed inset is provided in the lower right corner of the figure to highlight the differences among the grids. The coarse grid contains 0.71 M cells, the medium grid has 1.13 M cells, and the fine grid has 2.68 M cells. The maximum difference in C_p between coarse and medium grids is 1.3% and between the medium grid and the fine grid is 0.57%. Furthermore, Table I lists the grid sizes and the corresponding L/D obtained through RANS simulation, which shows that the difference between medium and fine grid is small enough. Hence, data from the medium grid of 1.13 M cells are reported here. The grid independence studies on the other geometries also yielded similar results, and all discussions are pertaining to the medium grid.

The dataset created to train the ANN model is generated by solving TM equations for M varying from 5 to 12 in steps of 0.25 and β corresponding to the cone angle varying from 5° to 45° in steps of 5°. Streamlines are chosen from X_s = 0.1 to 1 along the length of the cone. The streamlines so obtained are curve-fitted to polynomials, with the degree of the polynomial ranging from 1 to 7. Figure 16 shows the error in Y for a streamline at X_s = 0.1, M at 5.5, and a cone angle of 12°. The maximum percentage error in Y for the given range of (M, β, X_s) is found to be less than 8% (highest error for first order polynomial) with the maximum lying at the smallest values of (M, β, X_s) for all the polynomials. The error decreases with an increase in the degree of the polynomial. However, higher-degree polynomials involve the computation of many more coefficients, and hence, to strike a balance, the least degree of polynomial with a sufficiently accurate prediction is chosen. The percentage error in Y for the third order polynomial is less than 2% for all the values of inputs (M, β, X_s). Hence, to maintain a minimum number of outputs without losing the accuracy of the solution, a third-order polynomial is chosen in this study.

Figure 17 shows the influence of hyper parameters on the performance of the ANN model. It is observed that the RMSE (Root Mean Squared Error) for Bayesian regularization and Lavenberg–Marquardt model converges with an increase in the number of neurons. For 32 neurons in a layer, the RMSE of the models mentioned are 0.2465% and 0.2467%, respectively. Improvement in these values for two hidden layers is found to be minimal, and hence, a single-layered 32 neuron network with the Bayesian regularization model is chosen. Figure 18 pictorially depicts the streamlines obtained from the ANN model in comparison with the standard and curve-fitted streamlines. All three cases lie in close agreement with each other.

In order to indicate the effectiveness of the ANN model with respect to time consumption, the time taken to obtain the coordinates of 1 and 20 streamlines is tabulated in Table II. A system with 11th Gen Intel Core i7 processor is used to give the estimate of time for both cases. Evidently, the ANN model provides streamline coordinates of 1 and 20 streamlines ∼50 times faster and 20 times, respectively, when compared to the standard procedure of extracting streamline coordinates explained in Sec. II A.

Three waverider configurations with the same base curve specification are designed based on the different streamline tracing strategies mentioned in Sec. II. The standard waverider follows the TM solution, the simplified waverider considers geometrical relations, while the final waverider is designed by the novel ANN derived streamlines. The compression curve on the base plane for these three models are plotted in Fig. 19. The maximum percentage error with respect to the standard approach in the coordinates of ANN derived model is 0.68% and that of the simplified model is 8.47%.

The simplified conical waverider generation technique was introduced to reduce the computational complexity of the standard streamline tracing technique. In the simplified technique, the streamlines are taken parallel to the conical surface. However, it is well known that the conical flow streamlines are not parallel to the conical surface in the near field. Thus, this discrepancy leads to wide differences with the standard approach.

The ANN model is trained to follow the standard streamline tracing technique with a high degree of accuracy. Instead of solving the streamline differential equations at every point, the ANN model directly yields the equation of the streamline, thus being more computationally effective with better accuracy.

Figure 20 depicts the pressure plot on the base plane of the three comparative models with respect to freestream pressure. The shock angle in all the three cases is slightly higher than the angle predicted by the inviscid TM solution due to the presence of the boundary layer. The contours appear similar in the case of Figs. 20(a) and 20(c), while relatively high pressure is observed between the compression surface and shock of the simplified model in Fig. 20(b). This high pressure can be attributed to the increased shock angle perceived due to the lack of curvature in the streamlines of the simplified model. Significant spillage is noticed at the tip of the simplified model. The tip spillage is dramatically lower in the ANN derived model relative to the standard model. The effect of this spillage is evident through a drop in the aerodynamic efficiency of waveriders. The L/D of the standard and ANN derived model are noticeably close, as shown in Table III, while that of the simplified model drops by 12.52%.

In addition to L/D, Table III also presents the volumetric efficiencies of the models. If V is the internal volume and S is the planform area of the waverider, then its volumetric efficiency η is given by V^2/3/S. The planform area of all three models remains the same and is found to be 0.2089 m². While the difference in η of the standard and ANN derived model is negligible, a higher value for the simplified model is owed to the increase in its internal volume on considering streamlines parallel to the cone.

Streamline tracing from axisymmetric conical flow is an indispensable tool for generation of hypersonic waveriders and intakes critical to the development of hypersonic flight systems. The standard streamline tracing technique is computationally cumbersome, involving sequential solutions of the Taylor–Maccoll equations and then the differential equations for the streamlines. In the case of waverider geometry generation, a certain simplification was achieved through the simplified method, but the coordinates of the waverider geometry had significant differences due to the assumption of streamlines being parallel to the conical surface.

We have developed a novel ANN-based streamline tracing technique, which is both computationally efficient and highly accurate with respect to the standard approach. The ANN model is trained to output the coefficients of a third order polynomial fit to the streamlines, given the upstream Mach number, shock angle, and streamline originating location. The new ANN-based approach yields streamline coordinates within 0.68% difference from the standard approach and is about 20 times faster. The waverider generated from the ANN-based approach closely resembles the one from the standard approach and has much smaller spillage at the tips. The ANN based streamline tracing technique can be adapted to cases where the basic flow field is generated using 3D Euler or 3D RANS CFD simulations, which will yield a remarkable increase in the speed of geometry generation process, especially when several iterative geometries have to be computed in an optimization process.

The authors would like to thank the members of the Laboratory for Hypersonic and Shock Wave Research and Center of Excellence for Hypersonics, Indian Institute of Science, for their support and meaningful discussions.

The authors have no conflicts to disclose.

Anagha G. Rao: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Umesh Siddharth: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal). Srisha M. V. Rao: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.